53.18954pt t 0pt

Wavelets in Applied and Pure Mathematics

Vladimir V. Kisil

Abstract

The course gives an overview of wavelets (or coherent states) construction and its realisations in applied and pure mathematics. After a short introduction to wavelets based on the representation theory of groups we will consider:
The variety of applications is essentially grouped just around three groups: the Heisenberg group, SL2(R), and ax+b group.

Course Outline

Index

List of Figures

    1  The Gaussian function
    2  An example of Windowed Fourier Transform

Preface

The purpose of this course is to sketch in ten lectures a huge area related to wavelets. There are no precise boundaries of this area: it overlaps with many other subjects in pure mathematics and many applications in physics and engendering. Moreover there are many different approaches to wavelets based on rather different techniques. However this course is not intended to be complete and encyclopedic. Our main goal is to generate an interest in wavelets and to show that much of the theory and applications are related to groups and symmetries.
The word "wavelets" came to fashion about 15 years ago and is very popular now. On the other hand the notion of wavelets resemble coherent states used in quantum mechanics for 75 years already. Future analysis shows that many classic objects (e.g. from complex analysis) known at least from XIX century are essentially wavelets-coherent states too. This indicates that a significance of wavelets is above just a current fashion.
We apply the name "wavelets" to the whole range of related objects to stress their common origin and nature. Meanwhile the common usage of this term is much narrower. Objects called "wavelets" by us usually appear as coherent states (CS) in the literature. While commonly "wavelets" are coherent states related to ax+b group.

Lecture 1
What Are Wavelets and What Are They Good for?

In this introductory lecture one would only sketch an answer to the above question: even the whole course could not be enough for that. Now we just list several instances of wavelets appeared in different areas. All mentioned topics will be considered in greater details in following lectures.
Most of the listed facts should be well known to reader, we are just presenting them in a way highlighting the common structure. Similarities and differences of these instances of wavelets will be discussed in the final section  6.

1.1  Fourier Transform and Bases in Hilbert Spaces

We start from two basic examples which were at the beginning of harmonic and functional analysis.

1.1.1  Fourier Series and Basis in Hilbert Space

Consider the space L2[−π,π] of square integrable functions on [−π,π] with the Lebesgue measure. It is a Hilbert space with a scalar product
〈 f1,f2 〉 = 1
π
 π
⌠
⌡
−π 
 f1(x)
-
f
 
2 
 (x) dx.
(1.1)
Let us introduce the set of functions
e0(x)= 1
2
 , e2n(x)=cosnx,  e2n−11=sinnx ,    where     n ∈ N.
(1.2)
It is a straightforward calculation that
〈 ei,ej 〉=δi−j,
(1.3)
where δi−j is the Kronecker delta. Moreover the set (1.2) is a maximal family of functions in L2[−π,π] with the above property. In fact (1.3) could be taken as a definition of orthonormal base in a Hilbert space H.
It worths to state main properties of the family (1.2) in the generality of an arbitrary orthonormal base. For such a base ej the following is true [32,§ III.5.1]:

1.1.2  Fourier Transform

It is useful to compare the above properties of the Fourier series with the Fourier integral transform. The later is defined in L2(R) with the scalar product
〈 f1,f2 〉 = 1

√
2π
 ⌠
⌡

R 
 f1(t)
-
f
 
2 
 (t) dt.
(1.7)
by means of functions:
ea(t)=eiat,        a ∈ R.
(1.8)
A replacement of the orthonormal property (1.3) is the following identity (cf. (1.10)):
〈 ea,eb 〉=δ(a−b),
(1.9)
where δ(a−b) is the Dirac delta function and the identity is true in the sense of distributions [32,§ III.4.4].
The following is true [32,§ IV.2.3]:

1.2  Complex Analysis and Reproducing Kernels

We move to the classic Hilbert spaces in complex analysis which are examples of wavelets in pure mathematics. Particularly the first example named after G.H. Hardy, probably the purest mathematician of all times and nations.

1.2.1  The Hardy Space

Let H2(T) be the Hardy space of L2 functions on the unit circle T with an analytic continuation inside the unit disk D. The scalar product is defined as follows:
〈 f,f′ 〉= 1
2π
 ⌠
⌡

T 
 f(t)
-
f
 
 ′(t)  dt.
(1.11)
We could consider a set of functions in H2(T) parametrised by a point a of D:
ea(t)= 1
-
a
 
 eit − 1
 .
(1.12)
Then we could find similarly to cases of the Fourier series and integral that:

1.2.2  The Bergman Space

We consider the Bergman space in a way very similar to the Hardy space above. Let L2(D) be the space of square integrable function on D. There is a closed linear subspace-the Bergman space B2(D)-of analytic functions in L2(D). We define a family of functions
ea(z)= 1
(
-
a
 
 z−1)2
  ,        a ∈ D.
Remark 1 In both cases of the Hardy and the Bergman spaces we meet orthogonal projection P from the spaces of square integrable functions onto their subspace of analytic functions. Let Mb be a (bounded) operator on L2 of multiplication by a bounded function b. It is easy to see that for any such b the Töplitz operator Tb=PMb is a bounded operator on the subspace of analytical functions. We will link later such operators with the wavelet theory.

1.3  Qantum Mechanics and Quantisation

Now we turn to the object which combines the beauty of the mentioned above classic spaces of complex analysis and importance in applied area of quantum mechanics. As in case of the Fourier integral we start from L2(R) with the scalar product (1.7). Let us consider the family of functions:
ez(t) = e− (z2+t2)/2+√2zt,        t ∈ R,     z ∈ C.
Figure 1.1: The Gaussian function e−x2/2.
wavelets-gaussian.epsi.png
Note that e0(t)=e−t2/2 is the celebrated Gaussian shown on Figure 1. All other functions obtained from it by horizontal shifts and multiplication by a function eipt which takes value on the unit circle in C. In quantum mechanical language the function ez(t) with z=q+ip describes a state of a particle with an expectation of its coordinate equal to q, an expectation of its momentum-p, and the minimal value of product of coordinate and momentum dispersions [25,§ 1.3]. We will discuss a physical meaning in details letter on.
We again find a similar structure:

1.4  Signal Prosessing

wavelets-wt4-20.png
Figure 1.2: An example of Windowed Fourier Transform
The Fourier series and integral appeared as a tool for decomposition of an arbitrary oscillation (or signals) into a superposition of harmonic oscillations with a fixed frequencies. This technique is quite successful in the cases then spectrum of frequencies is independent from time or changes very slowly. But in many common situation like music, speech, etc. this is not true and the Fourier transformation is out of help.
To improve performance it is useful to introduce Windowed Fourier Transform (WFT ). It analyses the spectrum of frequence of not entire signal but only a part "seen" through a small windows. The position and size of the windows are among parameters of WFT. An example of such a transformation is shown on Figure 2 which is taken from the book [41], it is also instructional to view other pictures from this book on-line.
The word "wavelets" is commonly attributed to the area of signal processing. Decompositions of that type are of huge importance in signal processing and are under active investigation. We will discuss this topic in details due to course.

1.5  Functional Calculus

In the above consideration we oftenly meet a decomposition of an arbitrary function in to linear superposition of elementary ones, cf. (1.6), (2.6), (3.3). Because functions are used as models for operators such formulas could be employed for constructions of functional calculi. Particularly the Cauchy integral formula (2.2) inspires the Riesz-Dunford functional calculus defined by the integral formula:
f(A)= 1
2πi
 ⌠
⌡

T 
 f(t)
A−z
  dz,
(1.20)
for an operator A.

1.6  Discussion

The above consideration could rise many questions. We list now our answers to some of them:
The following lecture should give answer to more questions. But before we could proceed we will need a short overview of the representation theory.

Lecture 2
Groups and Homogeneous Spaces

The group theory and the representation theory are two enormous and interesting subjects themselves. However they are auxiliary in our consideration and we are forced to restrict ourselves only to brief and very dry overview.
Besides introduction to that areas presented in [42,56] we recommend additionally the books [31,55]. The representation theory intensively uses tools of functional analysis and on the other hand inspires its future development. We use the book [32] for references on functional analysis here and recommend it as a nice reading too.

2.1  Basics of Group Theory

We start from the definition of central object which formalizes the universal notion of symmetries.
Definition 1 A transformation group G is a nonvoid set of mappings of a certain set X into itself with the following properties:
  1. if g1 ∈ G and g2 ∈ G then g1g2 ∈ G;
  2. if g ∈ G then g−1 exists and belongs to G.
Exercise 2 List all transformation groups on a set of three elements.
Exercise 3 Verify that the following are groups in fact:
  1. Group of permutations of n elements;
  2. Group of n×n matrixes with non zero determinant over a field F under matrix multiplications;
  3. Group of rotations of the unit circle T;
  4. Groups of shifts of the real line R and plane R2;
  5. Group of linear fractional transformations of the extended complex plane.
Definition 4 An abstract group (or simply group) is a nonvoid set G on which there is a law of group multiplication (i.e. mapping G ×G→ G) with the properties
  1. associativity: g1(g2g3)=(g1g2)g3;
  2. the existence of identity: e ∈ G such that eg=ge=g for all g ∈ G;
  3. the existence of inverse: for every g ∈ G there exists g−1 ∈ G such that g g−1=g−1g=e.
Exercise 5 Check that any transformation group is an abstract group.
Exercise 6 Check that the following transformation groups (cf. Example 1.3) have the same law of multiplication, i.e. are equivalent as abstract groups:
  1. The group of isometric mapping of an equilateral triangle onto itself;
  2. The group of all permutations of a set of free elements;
  3. The group of invertible matrix of order 2 with coefficients in the field of integers modulo 2;
  4. The group of linear fractional transformations of the extended complex plane generated by the mappings z→ z−1 and z→ 1−z.
Exercise* 7 Expand the list in the above exercise.
It is simpler to study groups with the following additional property.
Definition 8 A group G is commutative if for all g1, g2 ∈ G, we have g1g2=g2g1.
However, most of interesting and important groups are noncommutative .
Exercise 9
  1. Which groups among found in Exercise 1.2 are commutative?
  2. Which groups among listed in Exercise 1.3 are noncommutative?
Groups could have some additional analytical structures, e.g. they could be a topological sets with a corresponding notion of limit . We always assume that our groups are locally compact  [31,§ 2.4].
Definition 10 If for a group G the group multiplication and the taking of inverse are continuous mappings then G is continuous group .
Even a better structure could be found among Lie groups  [31,§ 6], e.g. groups with a differentiable law of multiplication. Investigating such groups we could employ the whole arsenal of analytical tools, thereafter most of groups studied in this notes will be Lie groups.
Exercise 11 Check that the following are noncommutative Lie (and thus continuous) groups:
  1. [55,Chap. 7] The ax+b group : set of elements (a,b), a ∈ R+, b ∈ R with the group law:
    (a, b) * (a′, b′) = (aa′, ab′+b).
    The identity is (1,0), and (a,b)−1=(a−1,−b/a).
  2. The Heisenberg group  [26], [55,Chap. 1]: a set of triples of real numbers (s,x,y) with the group multiplication:
    (s,x,y)*(s′,x′,y′)=(s+s′+ 1
    2
    		 (x′y−xy′),x+x′,y+y′).
    		(2.1)
    The identity is (0,0,0), and (s,x,y)−1=(−s,−x,−y).
  3. The SL2(R) group [28,39]: a set of 2×2 matrixes with real entries a, b, c, d ∈ R, the determinant det =ad−bc equal to 1 and the group law coinciding with matrix multiplication:



    a
    b
    c
    d






    a′
    b′
    c′
    d′



    		=


    aa′+bc′
    ab′+bd′
    ca′+dc′
    cb′+dd′



    		.
    The identity is the unit matrix and



    a
    b
    c
    d



    		−1



     
    		 =


    d
    −b
    −c
    a



    		.
The above three groups are behind many important results of real and complex analysis [26,28,39] and we meet them many times in these notes.

2.2  Homogeneous Spaces and Invariant Measures

While abstract group are a suitable language for investigation of their general properties we meet groups in applications as transformation groups acting on a set X.
Let X be a set and let be defined an operation G: X→ X of G on X. There is an equivalence relation on X, say, x1 ∼ x2 ⇔ ∃g ∈ G: gx1=x2, with respect to which X is a disjoint union of distinct orbits [38,§ I.5].
Exercise 1 Let action of SL2(R) group on C by means of linear-fractional transformations :



a
b
c
d



: z → az+b
cz+d
 .
Show that there three orbits: the real axis R, upper (lower) half plane Rn±:
Rn±={ x±iy   |  x,y ∈ R, y > 0}.
Thus from now on, without lost of a generality, we assume that the operation of G on X is transitive , i. e. for every x ∈ X we have
Gx:=\mathrel
∪
g ∈ G
 
 g(x)=X.
In this case X is G- homogeneous space.
Exercise 2 Show that for any group G we could define its action on X=G as follows:
  1. The conjugation g: x → g x g−1 (which is even a group homomorphism, but is trivial for all commutative groups).
  2. The left shift λ(g): x → g x and the right shift ρ(g): x → x g−1.
If we fix a point x ∈ X then the set of elements Gx={g ∈ G  |  g(x)=x} obviously forms the isotropy (sub)group of x in G [38,§ I.5]. The set X is in the bijection with the factor set G/Gx for any x ∈ X.
Exercise 3 Find a subgroup which correspond to the given action of G on X:
  1. Action of ax+b group on R by the formula: (a,b): x → ax+b.
  2. Action of SL2(R) group on one of three orbit from Exercise 2.1.
To do some analysis on groups we need suitably defined basic operation: differentiation and integration . The first operation is naturally defined for Lie group. If G is a Lie group then the homogeneous space G/Gx is a smooth manifold (and a loop as an algebraic object) for every x ∈ X. Therefore the one-to-one mapping G/Gx → X: g→ g(x) induces a structure of C∞-manifold on S. Thus the class C0∞(X) of smooth functions with compact supports on x has the evident definition.
In order to perform an integration we need a suitable measure. A smooth measure dμ on X is called (left) invariant measure with respect to an operation of G on X if
⌠
⌡

X 
 f(x) dμ(x) = ⌠
⌡

X 
 f(g(x)) dμ(x),    for all g ∈ G,  f(x) ∈ C0∞(X).
(2.2)
Exercise 4 Show that measure y−2dy dx on the upper half plane R2+ is invariant under action from Exercise 2.1.
Left invariant measures on X=G is called the Haar measure . It always exists and is uniquely defined up to a scalar multiplier [55,§ 0.2]. An equivalent formulation of (2.1) is: G operates on L2(X,dμ) by unitary operators. We will transfer the Haar measure dμ from G to \mathfrakg via the exponential map exp:\mathfrakg→ G and will call it as the invariant measure on a Lie algebra \mathfrakg.
Exercise 5 Check that the following are Haar measures for corresponding groups:
  1. The Lebesgue measure dx on the real line R.
  2. The Lebesgue measure dφ on the unit circle T.
  3. dx/x is a Haar measure on the multiplicative group R+;
  4. dx dy/(x2+y2) is a Haar measure on the multiplicative group C\{0}, with coordinates z=x+iy.
  5. a−2 da db and a−1 da db are the left and right invariant measure on ax+b group .
  6. The Lebesgue measure ds dx dy of R3 for the Heisenberg group H1.
In this notes we assume all integrations on groups performed over the Haar measures.
Exercise 6 Show that invariant measure on a compact group G is finite and thus may be normalized to total measure 1.
The above simple result has surprisingly important consequences for representation theory of compact groups, which we state here without a proof.
Theorem 7 [31,§ 9.2]
  1. Every topologically irreducible representation of a compact group G is finite-dimensional and unitarizable.
  2. If T1 and T2 are two inequivalent irreducible representations, then every matrix element of T1 is orthogonal in L2(G) to every matrix element of T2.
  3. For a compact group G its dual space ∧G is discrete.
Definition 8 The left convolution f1*f2 of two functions f1(g) and f2(g) defined on a group G is
f1*f2(g)=⌠
⌡

G 
 f1(h) f2(h−1g) dh
Exercise 9 Let k(g) ∈ L1(G,dμ) and operator K on L1(G,dμ) is the left convolution operator with k, .i.e. K: f → k*f. Show that K commutes with all right shifts on G.
The following Lemma characterizes linear subspaces of L2(G,dμ) invariant under shifts in the term of ideals of convolution algebra L2(G,dμ) and is of the separate interest.
Lemma 10 A closed linear subspace H of L2(G,dμ) is invariant under left (right) shifts if and only if H is a left (right) ideal of the right group convolution algebra L2(G,dμ).
A closed linear subspace H of L2(G,dμ) is invariant under left (right) shifts if and only if H is a right (left) ideal of the left group convolution algebra L2(G,dμ).
PROOF. Of course we consider only the "right-invariance and right-convolution" case. Then the other three cases are analogous. Let H be a closed linear subspace of L2(G,dμ) invariant under right shifts and k(g) ∈ H. We will show the inclusion
[f*k]r(h)=⌠
⌡

G 
 f(g)k(hg) dμ(g) ∈ H,
(2.3)
for any f ∈ L2(G,dμ). Indeed, we can treat integral (2.2) as a limit of sums
N
∑
j=1 
 f(gj)k(hgj)∆j.
(2.4)
But the last sum is simply a linear combination of vectors k(hgj) ∈ H (by the invariance of H) with coefficients f(gj). Therefore sum (2.3) belongs to H and this is true for integral (2.2) by the closeness of H.
Otherwise, let H be a right ideal in the group convolution algebra L2(G,dμ) and let φj(g) ∈ L2(G,dμ) be an approximate unit of the algebra [21,§ 13.2], i. e. for any f ∈ L2(G,dμ) we have
[φj*f]r(h)=⌠
⌡

G 
 φj(g)f(hg) dμ(g) → f(h), when j→∞.
Then for k(g) ∈ H and for any h′ ∈ G the right convolution
[φj*k]r(hh′)=⌠
⌡

G 
 φj(g)k(hh′g) dμ(g) = ⌠
⌡

G 
 φj(h′−1g′)k(hg′) dμ(g′), g′=h′g,
from the first expression is tensing to k(hh′) and from the second one belongs to H (as a right ideal). Again the closeness of H implies k(hh′) ∈ H that proves the assertion. [¯]

Lecture 3
Elements of the Representation Theory

3.1  Representations of Groups

Objects unveil their nature in actions. Groups act on other sets by means of representations . A representation of a group G is a group homomorphism of G in a transformation group of a set. It is a fundamental observation that linear objects are easer to study. Therefore we begin from linear representations of groups.
Definition 1 A linear continuous representation of a group G is a continuous function T(g) on G with values in the group of non-degenerate linear continuous transformation in a linear space H (either finite or infinite dimensional) such that T(g) satisfies to the functional identity:
T(g1 g2) = T(g1) T(g2).
(3.1)
 Remark 2 If we have a representation of a group G by its action on a set X we can use the following linearization procedure . Let us consider a linear space L(X) of functions X→ C which may be restricted by some additional requirements (e.g. integrability, boundedness, continuity, etc.). There is a natural representation of G on L(X) which produced by its action on X:
g: f(x) → ρg f(x) = f(g·x),        where g ∈ G, x ∈ X.
(3.2)
Clearly this representation is already linear. However in many practical cases the formula for linearization (1.2) has some additional terms which are required to make it, for example, unitary.
Exercise 3 Show that T(g−1)=T−1(g) and T(e)=I, where I is the identity operator on B.
Exercise 4 Show that these are linear continuous representations of corresponding groups:
  1. Operators T(x) such that [T(x) f](t)=f(t+x) form a representation of R in L2(R).
  2. Operators T(n) such that T(n) ak=ak+n form a representation of Z in l2.
  3. Operators T(a,b) defined by
    [T(a,b) f](x) = √af(ax+b),        a ∈ R+,  b ∈ R
    		(3.3)
    form a representation of ax+b group in L2(R).
  4. Operators T(s,x,y) defined by
    [T(s,x,y) f] (t)=ei(2s−√2yt+xy) f(t− √2x)
    		(3.4)
    form Schrödinger representation of the Heisenberg group H1 in L2(R).
  5. Operators T(g) defined by
    [T(g) f](t) = 1
    ct+d
    		 f
    		 at+b
    ct+d

    		 ,     where g=


    a
    b
    c
    d



    		,
    		(3.5)
    form a representation of SL2(R) in L2(R).
In the sequel a representation always means linear continuous representation. T(g) is an exact representation (or faithful representation if T(g)=I only for g=e. The opposite case when T(g)=I for all g ∈ G is a trivial representation . The space H is representation space and in most cases will be a Hilber space [32,§ III.5]. If dimensionality of H is finite then T is a finite dimensional representation , in the opposite case it is infinite dimensional representation .
We denote the scalar product on H by 〈 ·,· 〉. Let {ej} be an (finite or infinite) orthonormal basis in H, i.e.
〈 ej,ej 〉=δjk,
where δjk is the Kroneker delta, and linear span of {ej} is dense in H.
Definition 5 The matrix elements tjk(g) of a representation T of a group G (with respect to a basis {ej} in H) are complex valued functions on G defined by
tjk(g) = 〈 T(g)ej,ek 〉.
(3.6)
 Exercise 6 Show that [56,§ 1.1.3]
  1. T(g) ek=∑j tjk(g) ej.
  2. tjk(g1g2)=∑n tjn(g1) tnk(g2).
It is typical mathematical questions to determine identical objects which may have a different appearance. For representations it is solved in the following definition.
Definition 7 Two representations T1 and T2 of the same group G in spaces H1 and H2 correspondingly are equivalent representations if there exist a linear operator A: H1 → H2 with the continuous inverse operator A−1 such that:
T2(g) = A T1(g) A−1,        ∀g ∈ G.
Exercise 8 Show that representation T(a,b) of ax+b group in L2(R) from Exercise 1.4iii is equivalent to the representation
[T1(a,b) f] (x) = ei[b/a]
√a
   f
 x
a

 .
(3.7)
 HINT. Use the Fourier transform. [¯]
The relation of equivalence is reflexive, symmetric, and transitive. Thus it splits the set of all representations of a group G into classes of equivalent representations. In the sequel we study group representations up to their equivalence classes only.
Exercise 9 Show that equivalent representations have the same matrix elements in appropriate basis.
Definition 10 Let T be a representation of a group G in H The adjoint representation T′(g) of G in H is defined by
T′(g)=( T(g−1))*,
where * denotes the adjoint operator in H.
Exercise 11 Show that
  1. T′ is indeed a representation.
  2. t′jk(g)=tkj(g−1).
Recall [32,§ III.5.2] that a bijection U: H → H is a unitary operator if
〈 Ux,Uy 〉=〈 x,y 〉,        ∀x, y ∈ H.
Exercise 12 Show that UU*=I.
Definition 13 T is a unitary representation of a group G in a space H if T(g) is a unitary operator for all g ∈ G. T1 and T2 are unitary equivalent representations if T1=UT2U−1 for a unitary operator U.
Exercise 14
  1. Show that all representations from Exercises 1.4 are unitary.
  2. Show that representations from Exercises 1.4iii and 1.8 are unitary equivalent.
HINT. Take that the Fourier transform is unitary for granted. [¯]
Exercise 15 Show that if a Lie group G is represented by unitary operators in H then its Lie algebra \mathfrakg is represented by self-adjoint (possibly unbounded) operators in H.
The following definition have a sense for finite dimensional representations.
Definition 16 A character of representation T is equal χ(g) = tr (T(g)), where tr  is the trace [32,§ III.5.2 (Probl.)] of operator.
Exercise 17 Show that
  1. Characters of a representation T are constant on the adjoint elements g−1hg, for all g ∈ G.
  2. Character is an algebra homomorphism from an algebra of representations with Kronecker's (tensor) multiplication [56,§ 1.9] to complex numbers.
HINT. Use that tr (AB)=tr (BA), tr (A+B)=tr A + tr B, and tr ( A ⊗B) = tr A tr B. [¯]
For infinite dimensional representation characters could be defined either as distributions [31,§ 11.2] or in infinitesimal terms of Lie algebras [31,§ 11.3].
The characters of a representation should not be confused with the following notion.
Definition 18 A character of a group G is a one-dimensional representation of G.
Exercise 19
  1. Let χ be a character of a group G. Show that a character of representation χ coincides with it and thus is a character of G.
  2. A matrix element of a group character χ coincides with χ.
  3. Let χ1 and χ1 be characters of a group G. Show that χ1 ⊗χ2 = χ1χ2 and χ′(g)=χ1(g−1) are again characters of G. In other words characters of a group form a group themselves.

3.2  Decomposition of Representations

The important part of any mathematical theory is classification theorems on structural properties of objects. Very well known examples are:
  1. The main theorem of arithmetics on unique representation an integer as a product of powers of prime numbers.
  2. Jordan's normal form of a matrix.
The similar structural results in the representation theory are very difficult. The easiest (but still rather difficult) questions are on classification of unitary representations up to unitary equivalence .
Definition 1 Let T be a representation of G in H. A linear subspace L ⊂ H is invariant subspace for T if for any x ∈ L and any g ∈ G the vector T(g)x again belong to L.
There are always two trivial invariant subspaces: the null space and entire H. All other are nontrivial invariant subspaces.
Definition 2 If there are only two trivial invariant subspaces then T is irreducible representation . Otherwise we have reducible representation .
For any nontrivial invariant subspace we could define the restriction of representation of T on it. In this way we obtain a subrepresentation of T.
Example 3 Let T(a), a ∈ R+ be defined as follows: [T(a)]f(x)=f(ax). Then spaces of even and odd functions are invariant.
Definition 4 If the closure of liner span of all vectors T(g) v is dense in H then v is called cyclic vector for T.
Exercise 5 Show that for an irreducible representation any non zero vector is cyclic.
The important property of unitary representation is complete reducibility.
Exercise 6 Let a unitary representation T has an invariant subspace L ⊂ H, then its orthogonal completion L⊥ is also invariant.
Theorem 7 [31,§ 8.4] Any unitary representation T of a locally compact group G could be decomposed in a (continuous) direct sum irreducible representations: T=∫X Tx  dμ(x).
The necessity of continuous sums appeared in very simple examples:
Exercise 8 Let T be a representation of R in L2(R) as follows: [T(a)f](x)=eiaxf(x). Show that
  1. Any measurable set E ⊂ R define an invariant subspace of functions vanishing outside E.
  2. T does not have invariant irreducible subrepresentations.
Definition 9 The set of equivalence classes of unitary irreducible representations of a group G is denoted by ∧G and called dual object (or dual space) of the group G.
Definition 10 A left regular representation Λ(g) of a group G is the representation by left shifts in the space L2(G) of square-integrable function on G with the left Haar measure
Λg: f(h) → f(g−1h).
(3.8)
The main problem of representation theory is to decompose a left regular representation Λ(g) into irreducible components.

3.3  Invariant Operators and Schur's Lemma

It is a pleasant feature of an abstract theory that we obtain important general statements from simple observations. Finiteness of invariant measure on a compact group is one such example. Another example is Schur's Lemma presented here.
To find different classes of representations we need to compare them each other. This is done by intertwining operators.
Definition 1 Let T1 and T2 are representations of a group G in a spaces H1 and H2 correspondingly. An operator A: H1 → H2 is called an intertwining operator if
A T1(g) = T2(g) A,        ∀g ∈ G.
If T1=T2=T then A is interntwinig operator or commuting operator for T.
Exercise 2 Let G, H, T(g), and A be as above. Show the following: [56,§ 1.3.1]
  1. Let x ∈ H be an eigenvector for A with eigenvalue λ. Then T(g)x for all g ∈ G are eigenvectors of A with the same eigenvalue λ.
  2. All eigenvectors of A with a fixed eigenvalue λ for a linear subspace invariant under all T(g), g ∈ G.
  3. If an operator A is commuting with irreducible representation T then A=λI.
HINT. Use the spectral decomposition of selfadjoint operators [32,§ V.2.2]. [¯]
The next result have very important applications.
Lemma 3 [Schur] [31,§ 8.2] If two representations T1 and T2 of a group G are irreducible, then every intertwining operator between them is either zero or invertible.
HINT. Consider subspaces kerA ⊂ H1 and im A ⊂ H2. [¯]
Exercise 4 Show that
  1. Two irreducible representations are either equivalent or disjunctive.
  2. All operators commuting with an irreducible representation form a field.
  3. Irreducible representation of commutative group are one-dimensional.
  4. If T is unitary irreducible representation in H and B(·,·) is a bounded semi linear form in H invariant under T: B(T(g)x,T(g)y)=B(x,y) then B(·,·)=λ〈 ·,· 〉.
HINT. Use that B(·,·)=〈 A·,· 〉 for some A [32,§ III.5.1]. [¯]

Lecture 4
Wavelets on Groups and Square Integrable Representations

A matured mathematical theory looks like a tree. There is a solid trunk which supports all branches and leaves but could not be alive without them. In the case of group approach to wavelets the trunk of the theory is a construction of wavelets from a square integrable representation [11], [2,Chap. 8]. We begin from this trunk which is a model for many different generalisations and will continue with some smaller "generalising" branches later.

4.1  Wavelet Transform on Groups

Let G be a group with a left Haar measure dμ and let ρ be a unitary irreducible representation of a group G by operators ρg, g ∈ G in a Hilbert space H.
Definition 1 Let us fix a vector w0 ∈ H. We call w0 ∈ B a vacuum vector or a mother wavelet (other less-used names are ground state , fiducial vector , etc.). We will say that set of vectors wg=ρ(g) w0, g ∈ G form a family of coherent states ( wavelets).
Exercise 2 If ρ is irreducible then wg, g ∈ G is a total set in H, i.e. the linear span of these vectors is dense in H.
The wavelet transform could be defined as a mapping from H to a space of functions over G via its representational coefficients
W: v →
^
v
 
 (g) = 〈 ρ(g−1)v,w0 〉 = 〈 v,ρ(g)w0 〉 = 〈 v,wg 〉.
(4.1)
Exercise 3 Show that the wavelet transform W is a continuous linear mapping and the image of a vector is a bounded continuous function on G. The liner space of all such images is denoted by W(G).
Exercise 4 Let a Hilbert space H has a basis ej, j ∈ Z and a unitary representation ρ of G=Z defined by ρ(k)ej=ej+k. Write a formula for wavelet transform with w0=e0 and characterise W(Z).
ANSWER. ∧v(n)=〈 v,en 〉. [¯]
Exercise 5 Let G be ax+b group and ρ is given by (cf. (1.3)):
[T(a,b) f](x) = 1
√a
 f
 x−b
a

 ,
(4.2)
in L2(R). Show that
  1. The representation is reducible and describe its irreducible components.
  2. for w0(x)=[1/(2πi (x+i))] coherent states are v(a,b)(x)=[√a/2πi (x−(b−ia))].
  3. Wavelet transform is given by
    ^
    v
     
    		 (a,b) = √a
    2πi
    		 ⌠
    ⌡

    R 
    		 v(x)
    x−(b+ia)
    		 dx,
    which resembles the Cauchy integral formula.
  4. Give a characteristic of W(G).
Proposition 6 The wavelet transform W intertwines ρ and the left regular representation Λ (2.1) of G:
W ρ(g) = Λ(g) W.
PROOF. We have:
[W( ρ(g) w)] (h)
=
〈 ρ(h−1) ρ(g) v , w0 〉
=
〈 ρ((g−1h)−1) v , w0 〉
=
[W v](g−1h)
=
[Λ(g) Wv] (h).
[¯]
Corollary 7 The function space W(G) is invariant under the representation Λ of G.
Wavelet transform maps vectors of H to functions on G. We can consider a map in the opposite direction sends a function on G to a vector in H.
Definition 8 The inverse wavelet transform M associated with a vector w′0 ∈ H maps L1(G) to H and is given by the formula:
M: L1(G) → H:
^
v
 
 (g) → M [
^
v
 
 (g)]
=
⌠
⌡

G 
^
v
 
 (g) w′g dμ(g)
=
⌠
⌡

G 
^
v
 
 (g) ρ(g) dμ(g) w′0,
(4.3)
where in the last formula the integral express an operator acting on vector w′0.
Exercise 9 Write inverse wavelet transforms for Exercises 1.4 and 1.5.
ANSWER.
  1. For Exercises 1.4: v=∑−∞∞ ∧v(n) en.
  2. For Exercises 1.5:
    v(x) = 1
    2πi
    		 ⌠
    ⌡

    R2+ 
    ^
    v
     
    		 (a,b)
    x−(b−ia)
    		 da db
    a[3/2]
    		 .
[¯]
Lemma 10 If the wavelet transform W and inverse wavelet transform M are defined by the same vector w0 then they are adjoint operators: W*=M.
PROOF. We have:
 
  		M
^
v
 
 ,wg  
 
=
 
  		⌠
⌡

G 
^
v
 
 (g′) wg′ dμ(g′),wg  
 
=
⌠
⌡

G 
^
v
 
 (g′) 〈 wg′,wg 〉 dμ(g′)
=
⌠
⌡

G 
^
v
 
 (g′)
〈 wg,wg′ 〉
 
  dμ(g′)
=
 
 
^
v
 
 ,W wg  
  		,
where the scalar product in the first line is on H and in the last line is on L2(G). Now the result follows from the totality of coherent states wg in H. [¯]
Proposition 11 The inverse wavelet transform M intertwines the representation Λ (2.1) on L2(G) and ρ on H:
M Λ(g) = ρ(g) M.
PROOF. We have:
M [Λ(g)
^
v
 
 (h)]
=
M [
^
v
 
 (g−1h)]
=
⌠
⌡

G 
^
v
 
 (g−1h) wg  dμ(h)
=
⌠
⌡

G 
^
v
 
 (h) w′gh′ dμ(h′)
=
ρ(g) ⌠
⌡

G 
^
v
 
 (h) w′h′ dμ(h′)
=
ρ(g) M [
^
v
 
 (h′)],
where h′=g−1h. [¯]
Corollary 12 The image M(L1(G)) ⊂ H of subspace under the inverse wavelet transform M is invariant under the representation ρ.
The following proposition explain the usage of the name "inverse" (not "adjoint" as it could be expected from Lemma 1.10) for M.
Theorem 13 The operator
P = M W: H → H
(4.4)
maps H into its linear subspace for which w′0 is cyclic. Particularly if ρ is an irreducible representation then P is cI for some constant c depending from w0 and w′0.
PROOF. It follows from Propositions 1.6 and 1.11 that operator MW: H → H intertwines ρ with itself. Then Corollaries 1.7 and 1.12 imply that the image MW is a ρ-invariant subspace of H containing w0. From irreducibility of ρ by Schur's Lemma  [31,§ 8.2] one concludes that MW=cI on C for a constant c ∈ C. [¯]
Remark 14 From Exercises 1.4 and 1.9 it follows that irreducibility of ρ is not necessary for MW=cI, it is sufficient that w0 and w′0 are cyclic only.
We have similarly
Theorem 15 Operator WM is up to a complex multiplier a projection of L1(G) to W(G).

4.2  Square Integrable Representations

So far our consideration of wavelets was mainly algebraic. Usually in analysis we wish that the wavelet transform could preserve an analytic structure, e.g. values of scalar product in Hilbert spaces. This accomplished if a representation ρ possesses the following property.
Definition 1 [31,§ 9.3] Let a group G with a left Haar measure dμ have a unitary representation ρ: G → L(H). A vector v ∈ H is called admissible vector if the function ∧v(g)=〈 ρ(g)w,w 〉 is non void and square integrable on G with respect to dμ:
0 < c2=⌠
⌡

G 
 〈 ρ(g)w,w 〉 〈 w,ρ(g)w 〉 dμ(g) < ∞.
(4.5)
If an admissible vector exists then ρ is a square integrable representation .
Square integrable representations of groups have many interesting properties (see [22,§ 14] for unimodular groups and [23], [2,Chap. 8] for not unimodular generalisation) which are crucial in the construction of wavelets. For example, for a square integrable representation all functions 〈 ρ(g)v1,v2 〉 with an admissible vector v1 and any v2 ∈ H are square integrable on G; such representation belong to dicrete series ; etc.
Exercise 2 Show that
  1. Admissible vectors form a linear space.
  2. For an irreducible ρ the set of admissible vectors is dense in H or empty.
HINT. The set of all admissible vectors is an ρ-invariant subspace of H. [¯]
Exercise 3
  1. Find a condition for a vector to be admissible for the representation (1.2) (and therefore the representation is square integrable).
  2. Show that w0(x)=[1/(2πi (x+i))] is not admissible.
For an admissible vector w we take its normalisation w0=[||w||/c] w to obtain:
⌠
⌡

G 
 | 〈 ρ(g)w0,w0 〉 |2 dμ(g) = ||w0||.
(4.6)
Such a w0 as a vacuum state produces many useful properties.
Proposition 4 If both wavelet transform W and inverse wavelet transform M for an irreducible square integrable representation ρ are defined by the same admissible vector w0 then the following three statements are equivalent:
  1. w0 satisfy (2.2);
  2. MW=I;
  3. for any vectors v1, v2 ∈ H:
    〈 v1,v2 〉=⌠
    ⌡

    G 
    ^
    v
     
    1 
    		 (g)
    ^
    v
     
    2 
    		 (g)
     
    		  dμ(g).
    		(4.7)
 PROOF. We already knew that MW=cI for a constant c ∈ C. Then (2.2) exactly says that c=1. Because W and M are adjoint operators it follows from MW=I on H that:
〈 v1,v2 〉 = 〈 MWv1,v2 〉 = 〈 Wv1,M*v2 〉=〈 Wv1,Wv2 〉,
which is exactly the isometry of W  (2.3). Finally condition (2.2) is a partticular case of general isometry of W for vector w0. [¯]
Exercise 5 Write the isometry conditions (2.3) for wavelet transforms for Z and ax+b groups (Exercises 1.4 and 1.5.
Wavelets from square integrable representation closely related to the following notion:
Definition 6 A reproducing kernel on a set X with a measure is a function K(x,y) such that:
K(x,x)
>
0,     ∀x ∈ X,
(4.8)
K(x,y)
=
K(y,x)
 
 ,
(4.9)
K(x,z)
=
⌠
⌡

X 
 K(x,y)K(y,z) dy.
(4.10)
 Proposition 7 The image W(G) of the wavelet transform W has a reproducing kernel K(g,g′)=〈 wg,wg′ 〉. The reproducing formula is in fact a convolution:
^
v
 
 (g′)
=
⌠
⌡

G 
 K(g′,g)
^
v
 
 (g) dμ(g)
=
⌠
⌡

G 
^
w
 
0 
 (g−1g′)
^
v
 
 (g) dμ(g)
(4.11)
with a wavelet transform of the vacuum vector ∧w0(g) = 〈 w0,ρ(g)w0 〉.
PROOF. Again we have a simple application of the previous formulas:
^
v
 
 (g′)
=
〈 ρ(g′−1)v,w0 〉
=
⌠
⌡

G 
 〈 ρ(h−1) ρ(g′−1) v,w0 〉 
〈 ρ(h−1) w0,w0 〉
 
  dμ(h)
(4.12)
=
⌠
⌡

G 
 〈 ρ((g′h)−1) v,w0 〉 〈 ρ(h) w0,w0 〉 dμ(h)
=
⌠
⌡

G 
^
v
 
 (g′h) 
^
w
 
0 
 (h−1)  dμ(h)
=
⌠
⌡

G 
^
v
 
 (g) 
^
w
 
0 
 (g−1g′) dμ(g),
where transformation (2.8) is due to  (2.3). [¯]
Exercise 8 Write reproducing kernels for wavelet transforms for Z and ax+b groups (Exercises 1.4 and 1.5.
Exercise* 9 Operator (2.7) of convolution with ∧w0 is an orthogonal projection of L2(G) onto W(G).
HINT. Use that an left invariant subspace of L2(G) is in fact an right ideal in convolution algebra, see Lemma 2.10. [¯]
Remark 10 To possess a reproducing kernel-is a well-known property of spaces of analytic functions. The space W(G) shares also another important property of analytic functions: it belongs to a kernel of a certain first order differential operator with Clifford coefficients (the Dirac operator) and a second order operator with scalar coefficients (the Laplace operator) [5,35,34,36], which we will consider that later too.
We consider only fundamentals of the wavelet construction here. There are much results which could be stated in an abstract level. To avoid repetition we will formulate it later on together with an interesting examples of applications.
The construction of wavelets from square integrable representations is general and straightforward. However we could not use it everywhere we may wish:
  1. Some important representations are not square integrable.
  2. Some groups, e.g. Hn, do not have square representations at all.
  3. Even if representation is square integrable, some important vacuum vectors are not admissible, e.g. w0(x)=[1/(2πi (x+i))] in 2.3ii.
  4. Sometimes we are interested in Banach spaces, while unitary square integrable representations are acting only on Hilbert spaces.
To be vivid the trunk of the wavelets theory should split into several branches adopted to particular cases and we describe some of them in the next lectures.

Lecture 5
Wavelets on Homogeneous Spaces: the Segal-Bargmann Space

We investigate a situation when a representation ρ of G is not square integrable in the sense of the previous Lecture but is square integrable modulo subgroup . An example which we use for illustration is the classic construction from quantum mechanics and is origin of coherent states.

5.1  Quantum Mechanical Setting

We begin from a statement of quantum problem [46,6] which could be naturally solved in the terms of wavelet transform. Mathematical formulation of quantum mechanics could be founded for example in [40], [32,§ V.3].
The states of a quantum mechanical system of n degrees of freedom (e.g. particle) are usually described by a function from the space H=L2(Rn). Depending on a physical interpretation it could be considered either as configuration space with real variables (q1, q2, …, qn) describing coordinates of the particle or momentum space with real variables (p1, p2, …, pn) describing its momenta. One could move from one description to another by the Fourier transform. Elements of H are also called wave functions.
The observables of the system are self-adjoint (possibly unbounded) operators on H. The result of measurement of an observable A on a state (wave function) φ with ||φ||=1 is a random distribution with an expectation 〈A 〉φ:
〈A 〉φ = 〈 Aφ,φ 〉
Exercise 1 Let we could find all expectations 〈A 〉φ for a fixed φ and any self-adjoint A. Show that we may calculate the random distributions of measurement.
HINT. Use expectation 〈 χ[a,b](A)φ,φ 〉, where χ[a,b](A) is a spectral projection of A on the interval [a,b] [32,§ V.1.3] to find a probability that the result of measurement of A on φ will be within interval [a,b]. [¯]
Among observables there is a special set of 2n primary ones: these are n observables of coordinates q1, ..., qn and n observables of momentum p1, ..., pn. All other observables usually could be expressed by means of primary ones: either as functions [40] or as wavelets [35]. The relation between primary observables are given by the Heisenberg commutation relations , i.e. the only non-trivial commutators among them are:
[qj,pk]=i(h/2p) δjkI.
(5.1)
Exercise 2 Check that the Heisenberg commutation relations (1.1) define a representation of the Lie Algebra of the Heisenberg group Hn .
Therefore a realisation of primary observables as self-adjoint operators H is connected with a unitary representation of the Heisenberg group in L2(Rn). We already met it: this is the Schrödinger representation (1.4):
[ρ(h/2p)(s,x,y) f] (q)=ei(2(h/2p) s−√{2(h/2p) } xq+(h/2p) xy) f(q−
√
 
2(h/2p)
 
 y).
(5.2)
(the representation (1.4) correspond to the case (h/2p) = 1).
The important result is the following theorem which asserts that we know all possible realisation of the Heisenberg commutation relations (1.1).
Theorem 3 [Stone-von Neumann] [31,§ 18.4], [55,§ 1.2] All unitary irreducible representations of the Heisenberg group Hn up to unitary equivalence are as follows
  1. For any (h/2p) ∈ (0,∞) the Schrödinger irreducible noncommutative unitary representations in L2(Rn)
    ρ±(h/2p)(s,x,y)=ei(±s·(h/2p) I ±x·(h/2p)1/2M +y·(h/2p)1/2D),
    		(5.3)
    where xM and yD are such unbounded self-adjoint operators on L2(Rn):
    (x ·(h/2p)1/2M)u(q)
    =
    (h/2p)1/2 ∑
    		xjqju(q),
    		(5.4)
    (y ·(h/2p)1/2D)u(q)
    =
    (h/2p)1/2
    i
    		 ∑
    		yj ∂u(q)
    ∂qj
    		 .
    		(5.5)
  2. For (q,p) ∈ R2n commutative one-dimensional representations on C:
    ρ(q,p)(s,x,y) u=ei(q x+p y)u, u ∈ C.
    		(5.6)
 Therefore there is essentially unique model for a quantum mechanical particle. Nevertheless it is worthwhile to look for some models which can act as alternatives for the Schrödinger representation. In particular, the Segal-Bargmann representation [6,46] serves to The huge abilities of the Segal-Bargmann (or Fock [24]) model are not yet completely employed, see for example new ideas in a recent paper [43].
Since the Segal-Bargmann model should give a representation Hn which is unitary equivalent to the Schrödinger one then it is naturally to construct an intertwining operator between them as a wavelet transform.

5.2  Fundamentals of Wavelets on Homogeneous Spaces

Let G be a group and G0 be its closed normal subgroup. Let X=G/G0 be the corresponding homogeneous space with a left invariant measure dμ. Let s: X → G be a Borel section in the principal bundle G → G/g0. Let ρ be a continuous representation of a group G by invertible unitary operators ρ(g), g ∈ G in a Hilbert space H.
For any g ∈ G there is a unique decomposition of the form g=s(x)h, h ∈ G0, x ∈ X. We will define r: G →G0: r(g)=h=(s−1(g))−1g from the previous equality and write a formal notation x=s−1(g). Then there is a geometric action of G on X → X defined as follows
g: x → g−1 ·x = s−1 (g−1 s(x)).
Example 1 As a subgroup G0 we select now the center of Hn consisting of elements (t,0). Of course X=G/G0 isomorphic to Cn and mapping s: Cn → G simply is defined as s(z)=(0,z). The Haar measure on Hn coincides with the standard Lebesgue measure on R2n+1 [55,§ 1.1] thus the invariant measure on X also coincides with the Lebesgue measure on Cn. Note also that composition law s−1(g· s(z)) reduces to Euclidean shifts on Cn. We also find s−1((s(z1))−1·s(z2))=z2−z1 and r((s(z1))−1·s(z2)) = [1/2] ℑz1z2.
Definition 2 Let G, G0, X=G/G0, s: X → G, ρ: G → L(H) be as above. We say that w0 ∈ H is a vacuum vector if it satisfies to the following two conditions:
ρ(h) w0 = χ(h) w0,        χ(h) ∈ C,  for all h ∈ G0;
(5.8)
⌠
⌡

X 
 | 〈 w0,ρ(s(x))w0 〉 |2 dx = ||w0||2.
(5.9)
We will say that set of vectors wx=ρ(x) w0, x ∈ X form a family of coherent states.
Note that mapping h → χ(h) from (2.1) defines a character of the subgroup G0. The condition (2.2) could be easily achieved by a renormalisation w0 as soon as we sure that the integral in the left hand side is finite.
Convention 3 In that follow we will usually write x ∈ X and x−1 ∈ X instead of s(x) ∈ G and s(x)−1 ∈ G correspondingly. The right meaning of "x" could be easily found from the context (whether an element of X or G is expected there).
Example 4 As a "vacuum vector" we will select the original vacuum vector of quantum mechanics-the Gauss function w0(q)=e−q2/2 (see Figure 1), which belongs to all L2(Rn). Its transformations are defined as follow:
wg(q)=[ρ(s,z) w0](q)
=
ei(2s−√2xq+xy) e−(q− √2y)2/2
=
e2is−(x2+y2)/2 e((x+iy)2−q2)/2−√2i(x+iy)q
=
e2is−zz/2e(z2−q2)/2−√2i z q.
Particularly [ρ(t,0) w0](q)=e−2itw0(q), i.e., it really is a vacuum vector in the sense of our definition with respect to G0.
Exercise 5 Check the square integrability condition  (2.2) for w0(q)=e−q2/2.
The wavelet transform (similarly to the group case ) could be defined as a mapping from G0 to a space of bounded continuous functions over G via representational coefficients
v →
^
v
 
 (g) = 〈 ρ(g−1)v,w0 〉 = 〈 v,ρ(g)*w0 〉.
Due to (2.1) such functions have simple transformation properties along orbits gG0, i.e. ∧v(gh)=χ(h)∧v(g), g ∈ G, h ∈ G0. Thus they are completely defined by their values indexed by points of X=G/G0. Therefore we prefer to consider so called reduced wavelet transform.
Definition 6 The reduced wavelet transform W from a Hilbert space G0 to a space of function W(X) on a homogeneous space X=G/G0 defined by a representation ρ of G on G0, a vacuum vector w0 is given by the formula
W: H → W(X): v →
^
v
 
 (x) = [Wv] (x)=〈 ρ(x−1) v,w0 〉 = 〈 v,ρ*(x)w0 〉.
(5.10)
 Example 7 The transformation (2.3) with the kernel [ρ(0,z) w0](q) is an embedding L2(Rn) → L2(Cn) and is given by the formula
^
f
 
 (z)
=
〈 f,ρs(z)f0 〉
=
π−n/4⌠
⌡

Rn 
 f(q) e−zz/2 e− (z2+q2)/2+√2zq dq
=
e−zz/2π−n/4⌠
⌡

Rn 
 f(q) e− (z2+q2)/2+√2zq dq .
(5.11)
Then ∧f(g) belongs to L2( Cn , dg) or its preferably to say that function \brevef(z)=ezz/2∧f(t0,z) belongs to space L2( Cn , e− | z |2 dg) because \brevef(z) is analytic in z. Such functions form the Segal-Bargmann space F2( Cn, e− | z |2 dg) of functions [6,46], which are analytic by z and square-integrable with respect to the Gaussian measure e− | z |2dz. We use notation \breveW for the mapping v → \brevev(z)=ezz/2Wv. Analyticity of \brevef(z) is equivalent to the condition ( [ ∂/( ∂zj )] + [1/2] zj I ) ∧f(z)=0 . The integral in (2.4) is the well-known Segal-Bargmann transform [6,46].
Exercise 8 Check that \brevew0(z)=1 for the vacuum vector w0(q)=e−q2/2.
There is a natural representation of G in W(X). It could be obtained if we first lift functions from X to G, apply the left regular representation Λ and then pul them back to X. The result defines a representation λ(g): W(X)→ W(X) as follow
[λ(g) f] (x) = χ(r(g−1·x)) f(g−1·x).
(5.12)
We recall that χ(h) is a character of G0 defined in (2.1) by the vacuum vector w0. Of course, for the case of trivial G0={e} (2.5) becomes the left regular representation Λ(g) of G.
Proposition 9 The reduced wavelet transform W intertwines ρ and the representation λ (2.5) on W(X):
W ρ(g) = λ(g) W.
 PROOF. We have with obvious adjustments in comparison with Proposition 1.6:
[W( ρ(g) v)] (x)
=
〈 ρ(x−1) ρ(g) v , w0 〉
=
〈 ρ((g−1s(x))−1) v , w0 〉
=
〈 ρ(r(g−1·x)−1)ρ(s(g−1·x)−1) v , w0 〉
=
〈 ρ(s(g−1·x)−1) v ,ρ*(r(g−1·x)−1) w0 〉
=
χ(r(g−1·x)−1) [W v] (g−1x)
=
λ(g) [Wv] (x).
[¯]
Corollary 10 The function space W(X) is invariant under the representation λ of G.
Example 11 Integral transformation (2.4) intertwines the Schrödinger representation (1.2) with the following realization of representation (2.5):
λ(s,z)
^
f
 
 (u)
=
^
f
 
0 
 (z−1·u)
-
χ
 
 (s+r(z−1·u))
=
^
f
 
0 
 (u−z)eis+iℑ(zu)
(5.13)
 Exercise 12
  1. Using relation \breveW=e−| z |2/2W derive from above that \breveW intertwines the Schrödinger representation with the following:
    \breveλ(s,z) \brevef(u) = \brevef0(u−z) e2is−zu−| z |2/2 .
  2. Show that infinitesimal generators of representation \breveλ are:
    ∂\breveλ(s,0,0)=iI,     ∂\breveλ(0,x,0)=−∂u−uI,     ∂\breveλ(0,0,y)=i(−∂z+zI)
We again introduce a transform adjoint to W.
Definition 13 The inverse wavelet transform M from W(X) to H is given by the formula:
M: W(X) → H:
^
v
 
 (x) → M [
^
v
 
 (x)]
=
⌠
⌡

X 
^
v
 
 (x) wx dμ(x)
=
⌠
⌡

X 
^
v
 
 (x) ρ(x) dμ(x) w0.
(5.14)
 Proposition 14 The inverse wavelet transform M intertwines the representation λ on W(X) and ρ on H:
M λ(g) = ρ(g) M.
PROOF. We have:
M [λ(g)
^
v
 
 (x)]
=
M [ χ(r(g−1·x))
^
v
 
 (g−1·x)]
=
⌠
⌡

X 
 χ(r(g−1·x))
^
v
 
 (g−1·x) wx  dμ(x)
=
χ(r(g−1·x)) ⌠
⌡

X 
^
v
 
 (x′) wg·x′ dμ(x′)
=
ρg ⌠
⌡

X 
^
v
 
 (x′) wx′ dμ(x′)
=
ρg M [
^
v
 
 (x′)],
where x′=g−1 ·x. [¯]
Corollary 15 The image M(W(X)) ⊂ H of subspace W(X) under the inverse wavelet transform M is invariant under the representation ρ.
Example 16 Inverse transformation to (2.4) is given by a realization of (2.7):
f(q)
=
⌠
⌡

Cn  
^
f
 
 (z) fs(z)(q) dz
=
⌠
⌡

Cn  
^
f
 
 (x,y) eiy(x−√2y) e−(q− √2y)2/2  dx dy
(5.15)
=
⌠
⌡

Cn  
 \brevef(z) e− (z2+q2)/2+√2zq  e− | z |2 dz.
The transformation  (2.8) intertwines the representations (2.6) and the Schrödinger representation (1.2) of the Heisenberg group.
The following proposition explain the usage of the name for M.
Theorem 17 The operator
P = M W: H → H
(5.16)
is a projection of H to its linear subspace for which w0 is cyclic. Particularly if ρ is an irreducible representation then the inverse wavelet transform M is a left inverse operator on H for the wavelet transform W:
MW=I.
 PROOF. It follows from Propositions 2.9 and 2.14 that operator MW: H → H intertwines ρ with itself. Then Corollaries 2.10 and 2.15 imply that the image MW is a ρ-invariant subspace of H containing w0. Because of MWw0=w0 we conclude that MW is a projection.
From irreducibility of ρ by Schur's Lemma [31,§ 8.2] one concludes that MW=cI on H for a constant c ∈ C. Particularly
MW w0 = ⌠
⌡

X 
 〈 ρ(x−1)w0,w0 〉 ρ(x) w0 dμ(x)=cw0.
From the condition (2.2) it follows that 〈 cw0,w0 〉=〈 MW w0,w0 〉=〈 w0,w0 〉 and therefore c=1. [¯]
We have similar
Theorem 18 Operator WM is a projection of L1(X) to W(X).
Corollary 19 In the space W(X) the strong convergence implies point-wise convergence.
PROOF. From the definition of the wavelet transform:

^
f
 
 (x)
=| 〈 f,ρ(x)w0 〉 | ≤ ||f|| ||w0||.
Since the wavelet transform is an isometry we conclude that | ∧f(x) | ≤ c||f|| for c=||w0||, which implies the assertion about two types of convergence. [¯]
Example 20 The corresponding operator for the Segal-Bargmann space P (2.9) is an identity operator L2(Rn) → L2(Rn) and (2.9) gives an integral presentation of the Dirac delta.
While the orthoprojection L2( Cn, e− | z |2 dg) → F2( Cn, e− | z |2 dg) is of a separate interest and is a principal ingredient in Berezin quantization [9,16]. We could easy find its kernel from (2.12). Indeed, ∧f0(z)=e − | z |2 , then the kernel is
K(z,w)
=
^
f
 
0 
 (z−1·w)
-
χ
 
 (r(z−1·w))
=
^
f
 
0 
 (w−z)eiℑ(zw)
=
exp
 1
2
 (− | w−z |2 +w
-
z
 
 −z
-
w
 
 )
=
exp
 1
2
 (− | z |2− | w |2) +w
-
z
 

 .
To receive the reproducing kernel for functions \brevef(z)=e| z |2 ∧f(z) in the Segal-Bargmann space we should multiply K(z,w) by e(−| z |2+ | w |2)/2 which gives the standard reproducing kernel = exp(− | z |2 +wz) [6,(1.10)].
We denote by W*: W*(X) → H and M*: H → W*(X) the adjoint (in the standard sense) operators to W and M respectively.
Corollary 21 We have the following identity:
〈 W v , M* l 〉 W(X) = 〈 v,l 〉H,        ∀v, l ∈ H,
(5.17)
or equivalently
⌠
⌡

X 
 〈 ρ(x−1) v,w0 〉 〈 ρ(x) w0,l 〉 dμ(x) = 〈 v,l 〉.
(5.18)
 PROOF. We show the equality in the first form (2.11) (but we will apply it often in the second one):
〈 W v , M* l 〉 W(X) = 〈 MW v ,l 〉H = 〈 v,l 〉H.
[¯]
Corollary 22 The space W(X) has the reproducing formula
^
v
 
 (y)=⌠
⌡

X 
^
v
 
 (x)  
^
b
 
0 
 (x−1·y) dμ(x),
(5.19)
where ∧b0(y)=[Ww0] (y) is the wavelet transform of the vacuum vector w0.
PROOF. Again we have a simple application of the previous formulas:
^
v
 
 (y)
=
〈 ρ(y−1)v,w0 〉
=
⌠
⌡

X 
 〈 ρ(x−1) ρ(y−1) v,w0 〉 〈 ρ(x) w0,w0 〉 dμ(x)
(5.20)
=
⌠
⌡

X 
 〈 ρ(s(y·x)−1) v,w0 〉 〈 ρ(x) w0,w0 〉 dμ(x)
=
⌠
⌡

X 
^
v
 
 (y·x) 
^
b
 
0 
 (x−1)  dμ(x)
=
⌠
⌡

X 
^
v
 
 (x) 
^
b
 
0 
 (x−1y) dμ(x),
where transformation (2.13) is due to (2.11). [¯]

5.3  Advanced Properties

We make the following simple but nice observation about the integral kernel of wavelet transform:
Proposition 1 Let ej, j ∈ N be an orthonormal basis in H, ∧ej be their images under a wavelet transform W then the kernel 〈 ·,ρ(x)w0 〉 of the wavelet transform W: v → 〈 x,ρ(x)w0 〉 has the following decomposition in the Dirac bra-ket notations:
〈 ·,ρ(x)w0 〉 = ∞
∑
j=1 
^
e
 
j 
 
  		〈 ej| = ∞
∑
j=1 
 〈 ·,ej 〉
^
e
 
j 
 .
Particularly if ej are orthogonal polynomials and ∧ej(x) are just powers of x then the kernel of the wavelet transform W is a generating function for ej.
Exercise 2 Give a proof.
Exercise 3
  1. Let Hn(q) be the Hermite polynomials , show that functions Hn(q)e−q2/2 form an orthonormal basis in L2(Rn).
  2. Show that functions zn/√{n!} are images under the Segal-Bargmann transform \breveW of functions Hn(q)e−q2/2. (Hint use that the Hermite polynomial obtained from the Gaussian by derivatives which are infinitesimals of the Schödinger representation of the Heisenberg group).
  3. Show that the Segal-Bargmann kernel is the generating function of Hermite polynomials .
Another example of this type is given by Bargmann in [6,§ 2g]. It links representations of SL2(R) in the Berman space and Laguerre polynomials, we will consider it later.
Proposition 4 Let A be an operator H→ H. Then the wavelet transform W intertwines A with an operator ∧A on W(X) given by the integral kernel ∧a(g,g′):
^
A
 
^
v
 
 (x)=⌠
⌡

X 
^
a
 
 (x,x′)
^
v
 
 (x′) dx′,     where
^
a
 
 (x,x′)=〈 Awx′,wx 〉.
(5.21)
 Example 5 [25,(1.81)] For the Segal-Bargmann space: if an operator A on L2(Rn) is given as an integral operator with a kernel a(q,q′) then ∧a(z,z′) is its double Segal-Bargmann transform.
Example 6 [Harmonic Oscillator] Let
H= 1
2
 n
∑
k=1 
 (p2k +q2k−1) = n
∑
k=1 
 a+k a−k,
be the Hamiltonian of a harmonic oscillator-the simplest non-trivial system in classic and quantum mechanics (creation a+ and annihilation a− operators were defined in (1.7)). The dynamics of quantum oscillator is governed by the Schrödinger equation :
d
dt
 φ(t) = iH φ(t),
and its solution φ(t)=eiHt φ(0) is given by means of the evolution operator eiHt. It is easier to construct the exponent (as any other function) of H if we could diagonalise H and that is done in the Segal-Bargmann representation. Indeed, H=∑k=1n zk [∂/(∂zk)]-the Euler operator. Its eigenvectors are zm with eigenvalues | m |. Consequently the evolution of the harmonic oscillator is given by
eiHt f(z)=f(eitz),
which is in a nice resemblance with geometrical dynamic of classic harmonic oscillator. In the contrast, the picture in L2(Rn) is not as simple. The eigenvectors of H are constructed from the Hermite polynomials (see Exercise 3.3) and dynamic is given by the complicated Mehler's formula [55,Chap. 1, (7.15)] .

Lecture 6
Wavelets in Banach Spaces and Functional Calculus

6.1  Coherent States for Banach Spaces

6.1.1  Abstract Nonsence

Let G be a group and G0 be its closed normal subgroup. Let X=G/G0 be the corresponding homogeneous space with an invariant measure dμ and s: X → G be a Borel section in the principal bundle G → G/G0. Let π be a continuous representation of a group G by invertible isometry operators ρg, g ∈ G in a (complex) Banach space B.
The following definition simulates ones from the Hilbert space case [1,§ 3.1].
Definition 1 Let G, G0, X=G/G0, s: X → G, π: G → L(B) be as above. We say that b0 ∈ B is a vacuum vector if for all h ∈ G0
ρ(h) b0 = χ(h) b0,        χ(h) ∈ C.
(6.1)
We will say that set of vectors bx=ρ(x) b0, x ∈ X form a family of coherent states if there exists a continuous non-zero linear functional l0 ∈ B* such that
  1. ||b0||=1, ||l0||=1, 〈 b0,l0 〉 ≠ 0;
  2. ρ(h)* l0=χ(h) l0, where ρ(h)* is the adjoint operator to ρ(h);
  3. The following equality holds
    ⌠
    ⌡

    X 
    		 〈 ρ(x−1) b0,l0 〉 〈 ρ(x) b0,l0 〉  dμ(x) = 〈 b0,l0 〉.
    		(6.2)
The functional l0 is called the test functional. According to the strong tradition we call the set (G,G0,π,B,b0,l0) admissible if it satisfies to the above conditions.
We note that mapping h → χ(h) from (1.1) defines a character of the subgroup G0. The following Lemma demonstrates that condition (1.2) could be relaxed.
Lemma 2 For the existence of a vacuum vector b0 and a test functional l0 it is sufficient that there exists a vector b0′ and continuous linear functional l′0 satisfying to (1.1) and 1.1ii correspondingly such that the constant
c = ⌠
⌡

X 
 〈 ρ(x−1)b′0,l′0 〉 〈 ρ(x) b′0,l′0 〉 dμ(x)
(6.3)
is non-zero and finite.
PROOF. There exist a x0 ∈ X such that 〈 ρ(x0−1) b0′,l′0 〉 ≠ 0 , otherwise one has c=0. Let b0=ρ(x−1) b0′||ρ(x−1) b0′||−1 and l0=l0′ ||l0′||−1. For such b0 and l0 we have 1.1i already fulfilled. To obtain (1.2) we change the measure dμ(x). Let c0=〈 b0,l0 〉 ≠ 0 then dμ′ = ||ρ(x−1) b0′|| ||l0′|| c0 c−1dμ is the desired measure. [¯]
Remark 3 Conditions (1.2) and (1.3) are known for unitary representations in Hilbert spaces as square integrability (with respect to a subgroup G0). Thus our definition describes an analog of square integrable representations for Banach spaces. Note that in Hilbert space case b0 and l0 are often the same function, thus condition 1.1ii is exactly (1.1). In the particular but still important case of trivial G0={e} (and thus X=G) all our results take simpler forms.
Convention 4 In that follow we will usually write x ∈ X and x−1 instead of s(x) ∈ G and s(x)−1 correspondingly. The right meaning of "x" could be easily found from the context (whether an element of X or G is expected there).
The wavelet transform (similarly to the Hilbert space case) could be defined as a mapping from B to a space of bounded continuous functions over G via representational coefficients
v →
^
v
 
 (g) = 〈 ρ(g−1)v,l0 〉 = 〈 v,π(g)*l0 〉.
Due to 1.1ii such functions have simple transformation properties along orbits gG0, i.e. ∧v(gh)=χ(h)∧v(g), g ∈ G, h ∈ G0. Thus they are completely defined by their values indexed by points of X=G/G0. Therefore we prefer to consider so-called reduced wavelet transform.
Definition 5 The reduced wavelet transform W from a Banach space B to a space of function F(X) on a homogeneous space X=G/G0 defined by a representation π of G on B, a vacuum vector b0 and a test functional l0 is given by the formula
W: B → F(X): v →
^
v
 
 (x) = [Wv] (x)=〈 ρ(x−1) v,l0 〉 = 〈 v,ρ*(x)l0 〉.
(6.4)
There is a natural representation of G in F(X). For any g ∈ G there is a unique decomposition of the form g=s(x)h, h ∈ G0, x ∈ X. We will define r: G → G0:r(g)=h=(s−1(g))−1g from the previous equality and write a formal notation x=s−1(g). Then there is a geometric action of G on X → X defined as follows
g: x → g−1 ·x = s−1 (g−1 s(x)).
We define a representation λ(g): F(X) →F(X) as follow
[λ(g) f] (x) = χ(r(g−1·x)) f(g−1·x).
(6.5)
We recall that χ(h) is a character of G0 defined in (1.1) by the vacuum vector b0. For the case of trivial G0={e} (1.5) becomes the left regular representation ρl(g) of G.
Proposition 6 The reduced wavelet transform W intertwines π and the representation λ (1.5) on F(X):
W ρ(g) = λ(g) W.
 PROOF. We have:
[W( ρ(g) v)] (x)
=
〈 ρ(x−1) ρ(g) v , l0 〉
=
〈 ρ((g−1s(x))−1) v , l0 〉
=
〈 ρ(r(g−1·x)−1)ρ(s(g−1·x)−1) v , l0 〉
=
〈 ρ(s(g−1·x)−1) v ,ρ*(r(g−1·x)−1) l0 〉
=
χ(r(g−1·x)−1) [W v] (g−1x)
=
λ(g) [Wv] (x).
[¯]
Corollary 7 The function space F(X) is invariant under the representation λ of G.
We will see that F(X) possesses many properties of the Hardy space. The duality between l0 and b0 generates a transform dual to W.
Definition 8 The inverse wavelet transform M from F(X) to B is given by the formula:
M: F(X) → B:
^
v
 
 (x) → M [
^
v
 
 (x)]
=
⌠
⌡

X 
^
v
 
 (x) bx dμ(x)
=
⌠
⌡

X 
^
v
 
 (x) ρ(x) dμ(x) b0.
(6.6)
 Proposition 9 The inverse wavelet transform M intertwines the representation λ on F(X) and π on B:
M λ(g) = ρ(g) M.
 PROOF. We have:
M [λ(g)
^
v
 
 (x)]
=
M [ χ(r(g−1·x))
^
v
 
 (g−1·x)]
=
⌠
⌡

X 
 χ(r(g−1·x))
^
v
 
 (g−1·x) bx  dμ(x)
=
χ(r(g−1·x)) ⌠
⌡

X 
^
v
 
 (x′) bg·x′ dμ(x′)
=
ρg ⌠
⌡

X 
^
v
 
 (x′) bx′ dμ(x′)
=
ρg M [
^
v
 
 (x′)],
where x′=g−1 ·x. [¯]
Corollary 10 The image M(F(X)) ⊂ B of subspace F(X) under the inverse wavelet transform M is invariant under the representation π.
The following proposition explain the usage of the name for M.
Theorem 11 The operator
P = M W: B → B
(6.7)
is a projection of B to its linear subspace for which b0 is cyclic. Particularly if π is an irreducible representation then the inverse wavelet transform M is a left inverse operator on B for the wavelet transform W:
MW=I.
 PROOF. It follows from Propositions 1.6 and 1.9 that operator MW: B → B intertwines π with itself. Then Corollaries 1.7 and 1.10 imply that the image MW is a π-invariant subspace of B containing b0. Because MWb0=b0 we conclude that MW is a projection.
From irreducibility of π by Schur's Lemma [31,§ 8.2] one concludes that MW=cI on B for a constant c ∈ C. Particularly
MW b0 = ⌠
⌡

X 
 〈 ρ(x−1)b0,l0 〉 ρ(x) b0 dμ(x)=cb0.
From the condition (1.2) it follows that 〈 cb0,l0 〉=〈 MW b0,l0 〉=〈 b0,l0 〉 and therefore c=1. [¯]
We have similar
Theorem 12 Operator WM is a projection of L1(X) to F(X).
We denote by W*: F*(X) → B* and M*: B* → F*(X) the adjoint (in the standard sense) operators to W and M respectively.
Corollary 13 We have the following identity:
〈 W v , M* l 〉 F(X) = 〈 v,l 〉B,        ∀v ∈ B,     l ∈ B*
(6.8)
or equivalently
⌠
⌡

X 
 〈 ρ(x−1) v,l0 〉 〈 ρ(x) b0,l 〉 dμ(x) = 〈 v,l 〉.
(6.9)
 PROOF. We show the equality in the first form (1.9) (but will apply it often in the second one):
〈 W v , M* l 〉 F(X) = 〈 MW v ,l 〉B = 〈 v,l 〉B.
[¯]
Corollary 14 The space F(X) has the reproducing formula
^
v
 
 (y)=⌠
⌡

X 
^
v
 
 (x)  
^
b
 
0 
 (x−1·y) dμ(x),
(6.10)
where ∧b0(y)=[Wb0] (y) is the wavelet transform of the vacuum vector b0.
PROOF. Again we have a simple application of the previous formulas:
^
v
 
 (y)
=
〈 ρ(y−1)v,l0 〉
=
⌠
⌡

X 
 〈 ρ(x−1) ρ(y−1) v,l0 〉 〈 ρ(x) b0,l0 〉 dμ(x)
(6.11)
=
⌠
⌡

X 
 〈 ρ(s(y·x)−1) v,l0 〉 〈 ρ(x) b0,l0 〉 dμ(x)
=
⌠
⌡

X 
^
v
 
 (y·x) 
^
b
 
0 
 (x−1)  dμ(x)
=
⌠
⌡

X 
^
v
 
 (x) 
^
b
 
0 
 (x−1y) dμ(x),
where transformation (1.11) is due to (1.9). [¯]
Remark 15 To possess a reproducing kernel-is a well-known property of spaces of analytic functions. The space F(X) shares also another important property of analytic functions: it belongs to a kernel of a certain first order differential operator with Clifford coefficients (the Dirac operator) and a second order operator with scalar coefficients (the Laplace operator) [5,35,34,36].
Let us now assume that there are two representations π′ and π" of the same group G in two different spaces B′ and B" such that two admissible sets (G,G0,π′,B′,b0′,l0′) and (G,G0,π",B",b0",l0") could be constructed for the same normal subgroup G0 ⊂ G.
Proposition 16 In the above situation if F′(X) ⊂ F"(X) then the composition T=M"W′ of the wavelet transform W′ for π′ and the inverse wavelet transform M" for π" is an intertwining operator between π′ and π":
Tπ′=π"T.
T is defined as follows
T: b → ⌠
⌡

X 
 〈 π′(x−1)b,l′0 〉  π"(x)b0" dμ(x).
(6.12)
This transformation defines a B"-valued linear functional (a distribution for function spaces) on B′.
The Proposition has an obvious proof. This simple result is a base for an alternative approach to functional calculus of operators [33,35] and will be used in Subsection 2.2. Note also that formulas (1.4) and (1.6) are particular cases of (1.12) because W and M intertwine π and λ.

6.1.2  Wavelets and a Positive Cone

The above results are true for wavelets in general. In applications a Banach space B is usually equipped with additional structures and wavelets are interplay with them. We consider an example of such interaction.
We recall [37], [30,Chap. X] the notion of positivity in Banach spaces. Let C ⊂ B be a sharp cone, i.e. x ∈ C implies that λx ∈ C and −λx ∉ C for λ > 0. We call such elements x ∈ C positive vectors. We say also that x ≥ y iff x−y is positive. There is the dual cone C* ⊂ B* defined by the condition
C*={f  |  f ∈ B*, 〈 b,f 〉 ≥ 0  ∀x ∈ C}.
An operator A:B→ B is called positive if Ab ≥ 0 for all b ≥ 0. If A is positive with respect to C then A* is positive with respect to C*.
Definition 17 We call a representation ρ(g) positive if there exists a vector b0 ∈ C such that ρ(x)b0 ∈ C for all x ∈ X. A linear functional f ∈ B* is positive (f > 0) with respect to a vacuum vector b0 if 〈 ρ(x)b0,f 〉 ≥ 0 for all x ∈ X and 〈 ρ(x)b0,f 〉 is not identically equal to 0.
Lemma 18 For any positive representation ρ(g) and vacuum vector b0 there exists a positive test functional.
PROOF. Obvious. [¯]
We consider an estimation of positive linear functionals.
Proposition 19 Let b ∈ B be a vector such that b=∫X ∧b(x) bx dμ(x). Let
  1. D(b)={〈 ρ(x−1)b,l0 〉   |  x ∈ X} be the set of value of reduced wavelets transform;
  2. \breveD(b) be a convex shell of the values of ∧b(x);
  3. ∧D(b)={ 〈 b,f 〉   |  f ∈ C*, ||f||=1, f ≥ 0 }.
Then
D(b) ⊂
^
D
 
 (b) ⊂ \breveD(b).
 PROOF. The first inclusion is obvious. The second could be easily checked:
〈 b,f 〉= 
  		⌠
⌡

X 
^
b
 
 (x) bx dμ(x),f  
  		= ⌠
⌡

X 
^
b
 
 (x) 〈 bx,f 〉 dμ(x).
[¯]

6.1.3  Singular Vacuum Vectors

In many important cases the above general scheme could not be carried out because the representation π of G is not square-integrable or even not square-integrable modulo a subgroup G0. Thereafter the vacuum vector b0 could not be selected within the original space B which the representation π acts on. The simplest mathematical example is the Fourier transform (see Example 3.1). In physics this is the well-known problem of absence of vacuum state in the constructive algebraic quantum field theory [47,48,49]. The absence of the vacuum within the linear space of system's states is another illustration to the old thesis Natura abhorret vacuum1 or even more specifically Natura abhorret vectorem vacui2.
We will present a modification of our construction which works in such a situation. For a singular vacuum vector the algebraic structure of group representations could not describe the situation alone and requires an essential assistance from analytical structures.
Definition 20 Let G, G0, X=G/G0, s: X → G, π: G → L(B) be as in Definition 1.1. We assume that there exist a topological linear space ∧B ⊃ B such that
  1. B is dense in ∧B (in topology of ∧B) and representation π could be uniquely extended to the continuous representation ∧π on ∧B.
  2. There exists b0 ∈ ∧B be such that for all h ∈ G0
    ^
    π
     
    		 (h) b0 = χ(h) b0,        χ(h) ∈ C.
    		(6.13)
  3. There exists a continuous non-zero linear functional l0 ∈ B* such that ρ(h)* l0=χ(h) l0, where ρ(h)* is the adjoint operator to ρ(h);
  4. The composition MW: B → ∧B of the wavelet transform (1.4) and the inverse wavelet transform (1.6) maps B to B.
  5. For a vector p0 ∈ B the following equality holds
     
          		⌠
    ⌡

    X 
    		 〈 ρ(x−1) p0,l0 〉 ρ(x) b0 dμ(x) ,l0  
          		= 〈 p0,l0 〉,
    		(6.14)
    where the integral converges in the weak topology of ∧B.
As before we call the set of vectors bx=ρ(x) b0, x ∈ X by coherent states; the vector b0-a vacuum vector; the functional l0 is called the test functional and finally p0 is the probe vector.
This Definition is more complicated than Definition 1.1. The equation (1.14) is a substitution for (1.2) if the linear functional l0 is not continuous in the topology of ∧B. Example 3.1 shows that the Definition does not describe an empty set. The function theory in R1,1 constructed in [34] provides a more exotic example of a singular vacuum vector.
We shall show that 1.20v could be satisfied by an adjustment of other components.
Lemma 21 For the existence of a vacuum vector b0, a test functional l0, and a probe vector p0 it is sufficient that there exists a vector b0′ and continuous linear functional l′0 satisfying to 1.20i-1.20iv and a vector p′0 ∈ B such that the constant
c= 
  		⌠
⌡

X 
 〈 ρ(x−1) p0,l0 〉 ρ(x) b0  dμ(x) ,l0  
 
is non-zero and finite.
The proof follows the path for Lemma 1.2. The following Proposition summarizes results which could be obtained in this case.
Proposition 22 Let the wavelet transform W (1.4), its inverse M (1.6), the representation λ(g) (1.5), and functional space F(X) be adjusted accordingly to Definition 1.20. Then
  1. W intertwines ρ(g) and λ(g) and the image of F(X)=W(B) is invariant under λ(g).
  2. M intertwines λ(g) and ∧π(g) and the image of M(F(B))=MW(B) ⊂ B is invariant under ρ(g).
  3. If M(F(X))=B (particularly if ρ(g) is irreducible) then MW=I otherwise MW is a projection B → M(F(X)). In both cases MW is an operator defined by integral
    b → ⌠
    ⌡

    X 
    		 〈 ρ(x−1) b,l0 〉 ρ(x) b0 dμ(x),
    		(6.15)
  4. Space F(X) has a reproducing formula
    ^
    v
     
    		 (y)= 
          		⌠
    ⌡

    X 
    ^
    v
     
    		 (x)  ρ(x−1y)b0  dx,l0  
     
    		(6.16)
    which could be rewritten as a singular convolution
    ^
    v
     
    		 (y)=⌠
    ⌡

    X 
    ^
    v
     
    		 (x)  
    ^
    b
     
    		 (x−1y) dx
    with a distribution b(y)=〈 ρ(y−1)b0,l0 〉 defined by (1.16).
The proof is algebraic and completely similar to Subsection 1.1.

6.2  Wavelets in Operator Algebras

We are going to apply the above abstract scheme to special spaces, which our main targets-wavelets on operator algebras. This gives a possibility to study operators by means of functions-symbols of operators.

6.2.1  Co- and Contravariant Symbols of Operators

We construct a realization of the wavelet transform as co- and contravariant symbols (also known as Wick and anti-Wick symbols) of operators. These symbols and their connections with wavelets in Hilbert spaces are known for a while [7,8,10,9]. However their realization (described bellow) as wavelets in Banach algebras seems to be new.
Let ρ(g) be a representation of a group G in a Banach space B by isometry operators. Then we could define two new representations for groups G and G ×G correspondingly in the space L(B) of bounded linear operators B→ B:
^
π
 
:
G → L(L(B)) : A → ρ(g)−1 A ρ(g),
(6.17)
~
π
 
:
G ×G → L(L(B)) : A → ρ(g1)−1 A ρ(g2),
(6.18)
where A ∈ L(B). Note that ∧π(g) are algebra automorphisms of L(B) for all g. Representation ~π(g1,g2) is an algebra homomorphism from L(B) to the algebra L(g1,g2)(B) of linear operators on B equipped with a composition
A1°A2 = A1 ρ(g1)−1ρ(g2) A2
with the usual multiplication of operators in the right-hand side. The rôle of such algebra homomorphisms in a symbolical calculus of operators was explained in [27]. It is also obvious that ∧π(g) is the restriction of ~π(g1,g2) to the diagonal of G ×G.
Let there are selected a vacuum vector b0 ∈ B and a test functional l0 ∈ B* for π. Then there are the canonically associated vacuum vector P0 ∈ L(B) and test functional f0 ∈ L*(B) defined as follows:
P0
:
B → B: b → P0b=〈 b,l0 〉b0;
(6.19)
f0
:
L(B) → C: A → 〈 Ab0,l0 〉.
(6.20)
They define the following coherent states and transformations of the test functional
Pg
=
^
π
 
 (g)P0=〈 ·,lg 〉bg,       P(g1,g2)=
~
π
 
 (g1,g2)P0=〈 ·,lg1 〉bg2,
fg
=
^
π
 
 *
 
 (g)f0=〈  · bg,lg 〉,       f(g1,g2)=
^
π
 
 *
 
 (g1,g2)f0=〈  · bg1,lg2 〉
where as usually we denote bg=ρ(g)b0, lg=ρ*(g)l0. All these formulas take simpler forms for Hilbert spaces if l0=b0.
Definition 1 The covariant (pre-)symbol A(x) (A(x1,x2)) of an operator A acting on a Banach space B defined by b0 ∈ B and l0 ∈ B* is its wavelet transform with respect to representation ∧π(g) (2.1) (~π(g1,g2) (2.2)) and the functional f0 (2.4), i.e. they are defined by the formulas
A(x)
=
(
^
π
 
 (x)A,f0) = 〈 ρ(x)−1Aρ(x) b0,l0 〉 = 〈 A bx,lx 〉,
(6.21)
A(x1,x2)
=
(
~
π
 
 (x1,x2)A,f0) = 〈 ρ(x1)−1Aρ(x2)b0,l0 〉 = 〈 A bx2,lx1 〉.
(6.22)
The contravariant (pre-)symbol of an operator A is a function \breveA(x) (a function \breveA(x1,x2) correspondingly) such that A is the inverse wavelet transform of \breveA(x) (of \breveA(x1,x2) correspondingly) with respect to ∧π(g) (~π(g1,g2)), i.e.
A
=
⌠
⌡

X 
 \breveA(x)
^
π
 
 (x) P0 dμ(x) = ⌠
⌡

X 
 \breveA(x) Px dμ(x),
(6.23)
A
=
⌠
⌡

X 
 ⌠
⌡

X 
 \breveA(x1,x2)
~
π
 
 (x1,x2) P0 dμ(x1) dμ(x2)
=
⌠
⌡

X 
 ⌠
⌡

X 
 \breveA(x1,x2) P(x1,x2) dμ(x1) dμ(x2),
(6.24)
where the integral is defined in the weak sense.
Obviously the covariant symbol \breveA(x) is the restriction of the covariant presymbol \breveA(x1,x2) to the diagonal of G×G.
Proposition 2 A mapping σ: A → σA(x1,x2) of operators to their covariant symbols is the algebra homomorphism from algebra of operators on B to algebra of integral operators on F(G), i.e.
σA1A2(x1,x3)=⌠
⌡

X 
 σA1(x1,x2) σA2(x2,x3) dμ(x2).
(6.25)
 PROOF. One could easily see that:
⌠
⌡

X 
 σA1(x1,x2) σA2(x2,x3) dμ(x2)       
=
⌠
⌡

X 
 〈 ρ(x1)A1ρ(x2−1) b0,l0 〉 〈 ρ(x2)A2ρ(x3−1) b0,l0 〉 dμ(x2)
=
⌠
⌡

X 
 〈 ρ(x2−1) b0,A1*ρ*(x1)l0 〉 〈 ρ(x2)A2ρ(x3−1) b0,l0 〉 dμ(x2)
=
〈 A2ρ(x3−1) b0,A1*ρ*(x1)l0 〉
(6.26)
=
〈 ρ(x1)A1A2ρ(x3−1) b0,l0 〉
=
σA1A2(x1,x3),
where transformation (2.10) is due to (1.9). [¯]
The following proposition is obvious.
Proposition 3 An operator A could be reconstructed from its covariant presymbol A(g1,g2) by the formula
Av=⌠
⌡

G 
 ⌠
⌡

G 
 A(g1,g2)
^
v
 
 (g2) dμ(g2) bg1 dμ(g1).
We have a particular interest in operators closely connected with the representation ρg.
Proposition 4 Let an operator A on B is defined by the formula
Av=⌠
⌡

G 
 a(g) ρg v dμ(g)
for a function a(g) on G. Then A′W = W A where A′ is a two-sided convolution on G defined by the formula
[A′
^
v
 
 ] (h) = ⌠
⌡

G 
 ⌠
⌡

G 
 a(g1)
^
b
 
0 
 (g2)
^
v
 
 (g1−1 h g2) dμ(g1) dμ(g2).
For operator algebras there are the standard notions of positivity: any operator of the form A*A is positive; if algebra is realized as operators on a Hilbert space H then b ∈ H defines a positive functional fb(A)=〈 Ab,b 〉. Thus the following proposition is a direct consequence of the Proposition 1.19.
Proposition 5 [7,Thm. 1] Let A be an operator, let D(A) be the set of values of the covariant symbol A(x), let \breveD(A) be a convex shell of the values of contravariant symbol \breveA(x). Let ∧D(A) be the set of values of the quadratic form 〈 Ab,b 〉 for all vectors ||b||=1. Then
D(b) ⊂
^
D
 
 (b) ⊂ \breveD(b).
 Example 6 There are at least two very important realizations of symbolical calculus of operators. The theory of pseudodifferential operators (PDO) [18,50,54] is based on the Schrödinger representation of the Heisenberg group Hn (see Subsection 3.1) on the spaces of functions Lp(Rn) [27]. The Wick and anti-Wick symbolical calculi [7,9] arise from the Segal-Bargmann representation [46,6] (see Subsection 2.1) of the same group Hn. Connections (intertwining operators) between these two representations were exploited in [27] to obtain fundamentals of the theory of PDO.

6.2.2  Functional Calculus and Group Representations

This Subsection illustrates a new approach to functional calculus of operators outlined in [33,35]. The approach uses the intertwining property for two representations instead of an algebraic homomorphism.
Let \mathfrakB be a Banach algebra and T ⊂ \mathfrakB be its subset of elements. Let G be a group, G0 be its normal subgroup and X=G/G0-the corresponding homogeneous space. We assume that there is a representation τ depending from T ⊂ \mathfrakB defined on measurable functions from L(X,\mathfrakB) by the formula
τ(g) f(x) = t(g,x) f(g−1·x),        f(x) ∈ L(X,\mathfrakB),
(6.27)
where t(g,x): \mathfrakB → \mathfrakB depends from x ∈ X and g ∈ G. It is convenient to use a linear functional l ∈ \mathfrakB′ to make the situation more tractable by reducing it to the scalar case. Using l we could define a representation τl(x) on F(X) by the following formula
τl(x): fl(y)=〈 f(y),l 〉 → [τl(x)fl](y)=〈 τ(x) f(y),l 〉,
(6.28)
where f(y) ∈ L(X,\mathfrakB), l ∈ \mathfrakB′. We will understand convergence of all integrals involving τ in a weak sense, i.e. as convergence of all corresponding integrals with τl, l ∈ \mathfrakB′. We also say that τ is irreducible if all τl are irreducible.
Remark 7 If \mathfrakB is realized as an algebra of operators on a Banach space B then l ∈ \mathfrakB′ could be realized as an element of B⊗B′. In this case the formula (2.12) looks like (2.5). The important difference is the following. In (2.5) the representation in the operator algebra \mathfrakB arises from a representation in Banach space B and is the same for all elements of \mathfrakB. Representation τl in (2.12) is defined via the representation τ which depends in its turn from a set T ⊂ \mathfrakB. Such representations are usually connected with some (non-linear) geometric actions of a group directly on operator algebra. Examples of these geometric actions are the representation of the Heisenberg group (3.10) leading to the Weyl functional calculus, fractional-linear transformations of operators leading [35] to Dunford-Riesz functional calculus and monogenic functional calculus [33]. Thus such representations contain important information on T.
We also assume that there is a representation π of G in F(X) with a vacuum vector b0, a test functional l0 and the system of wavelets (coherent states) bx, x ∈ X, which were main actors in the previous Section. Let ρ*(g)=ρ(g−1)* be the adjoint representation of ρ(g) in F′(X).
We need a preselected element T0(x) ∈ L(X,\mathfrakB) which plays a rôle of a vacum vector for the representation τ, it is defined by the condition:
⌠
⌡

X 
^
b
 
0 
 (x′) τ(x′) T0(x) dx′=T0(x),
(6.29)
where ∧b0(x)=〈 ρ(x−1) b0,l0 〉 is the wavelet transform of the vacuum vector b0 ∈ F(X) for π.
Lemma 8 A vacuum vector for τ always exists and is given by the formula
T0(x)=⌠
⌡

X 
^
b
 
0 
 (x′)  τ(x′) T(x) dx′,
(6.30)
where T(x) ∈ L(X,\mathfrakB) is an arbitrary element which the integral (2.14) converges for. If τ is irreducible (i.e. all τl (2.12) are irreducible) in the linear span of τ(x′)T(x), x′ ∈ X then T0(x) does not depend from a particular chose of T(x).
PROOF. First we could easily verify condition (2.13) for the T0 defined by (2.14):
⌠
⌡

X 
^
b
 
0 
 (x) τ(x) T0(x1) dx = ⌠
⌡

X 
^
b
 
0 
 (x) τ(x) ⌠
⌡

X 
^
b
 
0 
 (x′)  τ(x′) T(x1) dx′ dx                        
=
⌠
⌡

X 
 ⌠
⌡

X 
^
b
 
0 
 (x) 
^
b
 
0 
 (x′)   τ(x)τ(x′) T(x1) dx′ dx
=
⌠
⌡

X 
⌠
⌡

X 
^
b
 
0 
 (x)
^
b
 
0 
 (x−1x")  dx
 τ(x") T(x1) dx"   
(6.31)
=
⌠
⌡

X 
^
b
 
0 
 (x") τ(x") T(x1) dx"
(6.32)
=
T0(x1).
Here we use the change of variables x"=x·x′ in (2.15) and reproducing property (1.10) of ∧b0(x) in (2.16).
To prove that for any admissible T(x) we will receive the same T0(x) is enough to pass from the representation τ to representations τl (2.12) defined by l ∈ \mathfrakB′. Then we deal with scalar valued (not operator valued) functions and knew that one could use any admissible vector Tl(x)=〈 T(x),l 〉 as a vacuum vector in the reconstruction formula (1.6). [¯]
Now we could specify the Definition 1.1 from [33] as follows.
Definition 9 Let G, G0, X=G/G0, \mathfrakB, T, τ, π, F, b0, T0 be as described above. One says that a continuous linear \mathfrakB-valued functional ΦT(·,x): F (X) → \mathfrakB, parametrized by a point x ∈ X and depending from T ⊂ \mathfrakB:
ΦT(·,x): f(y) → [ΦTf](x) = ⌠
⌡

X 
 f(y) ΦT (y,x) dy
is a functional calculus if
  1. ΦT is an intertwining operator between ρ(g) and τ(g), namely
    [ΦTρ(g)f(y)](x) = τ(g)[ΦT f(y)](x),
    		(6.33)
    for all g ∈ G and f(y) ∈ F(X)
  2. ΦT maps the vacuum vector b0(y) for the representation π to the vacuum vector T0(x) for the representation τ:
    [ΦTb0(y)] (x)=T0(x).
    		(6.34)
\mathfrakB-valued distribution ΦT(y, x0), s(x0)=e ∈ G associated with \mathfrakB-valued linear functional on F(X) is called a spectral decomposition of operators T.
Representation τ in (2.17) is defined by (2.11):
τ(g) [ΦT f(y)]( x) = t(g,x) [ΦT f(y)](g−1·x).
We could state (2.17) equivalently as
[Iy ⊗τx(g)] Φ(y,x) = [ρy*(g−1) ⊗Ix] Φ(y,x).
Remark 10 The functional calculus ΦT(y,x) as defined here has the explicit covariant property with respect to variable x. Thus it could be restored by the representation τ from a single value, e.g. ΦT(y, s−1(e)), where e is the identity of G. We particularly will calculate only [ΦTf](s−1(e)) in Subsection 3.3 as the value of a functional calculus. This value is usually denoted by f(T) and is exactly the functional calculus of operators in the traditional meaning.
In particular cases different characteristics of the spectral decomposition could give relevant information on the set of operators T, e.g. the support supp yΦT(y,x0) of ΦT(y,x0)
f(y)=0 ∀y ∈ supp yΦT(y,x0)     ⇒    [ΦTf](x0)=0
is called (joint) spectrum of set T ⊂ \mathfrakB. This definition of the spectrum is connected with the Arveson-Connes spectral theory [4,17,52] while there are several important differences mentioned in [33,Rem. 4.4].
In the paper [33] the approach was illustrated by a newly developed functional calculus for several non-commuting operators based on Möbius transformations of the unit ball in Rn. It was shown in [35,§ 7] that the classic Dunford-Riesz functional calculus is generated by a representation of SL(2,R) within this procedure. However an abstract scheme of the approach was not presented yet. We give some its elements here.
From Proposition 1.16 we know a general form of an intertwining operator of two related representations of a group, which could be employed here. Let l0(y) be the distribution corresponding to a test functional l0 for the representation π on F(X) such that we could write
〈 f(y),l0 〉F(X) = ⌠
⌡

X 
 f(y) l0(y) dy.
We denote also by ρ*(x) l0(y), x ∈ X, y ∈ X distributions corresponding to linear functionals ρ*(x) l0, where ρ*(x) is the adjoint representation to π on the space F(X).
Proposition 11 [Spectral syntesis] Under assumption of Proposition 1.16 the functional calculus exists and is unique. The spectral decomposition ΦT(y,x) as a distribution on X is given by the formula
ΦT(y,x)=⌠
⌡

X 
 ρ*(x) l0(y)  τ(x) T0(x)  dx.
(6.35)
The functional calculus ΦT(·,x) as a mapping F(X) → \mathfrakB is given correspondingly
ΦT(·,x): f(y) → [ΦTf(y)](x) = ⌠
⌡

X 
 ⌠
⌡

X 
 〈 f(y),ρ*(x′) l0(y) 〉 τ(x′) T0(x) dy  dx′.
(6.36)
 PROOF. Obviously (2.19) and (2.20) are equivalent. Thus we will prove (2.20) only. For an arbitrary f(y) ∈ F(X) we could write
[ΦT f(y)](x)
=

ΦT ⌠
⌡

X 
^
f
 
 (x′) ρ(x′) b0(y) dx′
 (x)
(6.37)
=
⌠
⌡

X 
^
f
 
 (x′) [ΦT ρ(x′) b0(y)](x) dx′
(6.38)
=
⌠
⌡

X 
^
f
 
 (x′) τ(x′) [ΦT b0(y)](x) dx′
(6.39)
=
⌠
⌡

X 
^
f
 
 (x′) τ(x′) T0(x) dx′
(6.40)
=
⌠
⌡

X 
 〈 f(y),ρ*(x′) l0 〉 τ(x′) T0(x) dx′
(6.41)
We use in (2.21) that functions in F(X) are superpositions of coherent states, transformation (2.22) is made by linearity and continuity of ΦT, step (2.23) is due to condition (2.17) and we finally apply (2.18) to receive (2.24). Thus it is proven that the functional calculus which is continuous, linear, and satisfies to (2.17) and (2.18) (if exists) is unique and given by (2.25). Now we should check that (2.25) really gives the right answer.
We will check first that (2.25) satisfies to (2.17):
τ(g) Φ(y,x)
=
τ(x1) ⌠
⌡

X 
 ρ*(x′) l0(y) τ(x′) T0(x) dx′
=
⌠
⌡

X 
 ρ*(x′) l0(y) τ(g·x′) T0(x) dx′
=
⌠
⌡

X 
 ρ*(g−1·x") l0(y) τ(x") T0(x) dx"
(6.42)
=
ρ*(g−1) ⌠
⌡

X 
 ρ*(x") l0(y) τ(x") T0(x) dx"
=
ρ*(g−1) Φ(y,x) ,
where we made substitution x"=g·x′ in (2.26). Finally (2.18) directly follows from the condition (2.13). [¯]
Let there exists L0(x) ∈ L′(X,\mathfrakB)-a test functional for a vacuum vector T0(x) and representation τ, i.e.
〈 T0,L0 〉L(X,\mathfrakB) = ⌠
⌡

X 
 〈 τ(x−1 T0),L0 〉L(X,\mathfrakB) 〈 τ(x) T0,L0 〉L(X,\mathfrakB) dx,
where
〈 T0,L0 〉L(X,\mathfrakB) = ⌠
⌡

X 
 〈 T0(x),L0(x) 〉\mathfrakB dx
and 〈 T0(x),L0(x) 〉\mathfrakB is the pairing between \mathfrakB and \mathfrakB′.
Proposition 12 [Spectral analysis] If a \mathfrakB-valued function F(x) from L(X,\mathfrakB) belongs to the closer of the linear span of τ(x′)T0(x), x′ ∈ X then
F(x) = [ΦT f(y)] (x),
where
f(y) = ⌠
⌡

X 
 〈 τ(x−1F),L0 〉L(X,\mathfrakB)  ρ(x)b0(y) dx.
(6.43)
 PROOF. The formula (2.27) is just another realization of intertwining operator (1.12) from Proposition 1.16. [¯]
Let K: F(X1) → F(X2) be an intertwining mapping between two representations ρ1 and ρ2 of groups G1 and G2 in spaces F(X1) and F(X2) respectively. Let K(z,y), z ∈ X1, y ∈ X2 be the Schwartz kernel of K.
Theorem 13 [Mapping of spectral decompositions] Let
f1(z) = [Kf2](z)
=
⌠
⌡

X2 
 f2(y) K(z,y) dy
ΦT1 (z,x)
=
⌠
⌡

X2 
 K(z,y) ΦT2(y,x) dy,
where a functional calculus ΦT2 is defined by representations ρ2 and τ. Then ΦT2(z,x) is a functional calculus for ρ1 and τ and we have an identity:
[ΦT1 f1(z)](x) = [ΦT2 f2(y)] (x).
(6.44)
 PROOF. The intertwining property for ΦT2(z,x) follows from transitivity. The identity (2.28) is a simple application of the Fubini theorem:
[ΦS g(z)](x)
=
⌠
⌡

X1 
 g(z) ΦS (z,x) dz
=
⌠
⌡

X1 
 g(z) ⌠
⌡

X2 
 K(z,y) ΦT(y,x)  dy dz
=
⌠
⌡

X2 
 ⌠
⌡

X1 
 g(z) K(z,y)  dz ΦT(y,x)  dy
=
⌠
⌡

X2 
 g(f(y)) ΦT(y,x)  dy
=
[ΦT g(f(y))](x).
[¯]
This Theorem could be turned in the spectral mapping theorem under suitable conditions [33,Thm. 3.19].

6.3  Examples

We are going to demonstrate that the above construction is not only algebraically attractive but also belongs to the heart of analysis. More examples could be found in [15,35,34] and will be given elsewhere.

6.3.1  The Heisenberg Group and Schrödinger Representation

We will consider a realization of the previous results in a particular cases of the Fourier transform and Segal-Bargmann [6,46] type spaces Fp(Cn). They arise from representations of the Heisenberg group Hn [25,26,55] on Lp(Rn).
The Lie algebra \mathfrakhn of Hn spanned by {T,Pj, Qj}, n=1,…,n is defined by the commutation relations:
[Pi,Qj]=Tδij.
(6.45)
They are known from quantum mechanics as the canonical commutation relations of coordinates and momentum operators. An element g ∈ Hn could be represented as g=(t,z) with t ∈ R, z=(z1,…,zn) ∈ Cn and the group law is given by
g*g′=(t,z)*(t′,z′)=(t+t′+ 1
2
 n
∑
j=1 
 ℑ(
-
z
 
j 
 zj′), z+z′),
(6.46)
where ℑz denotes the imaginary part of a complex number z. The Heisenberg group is (non-commutative) nilpotent step 2 Lie group.
We take a representation of Hn in Lp(Rn), 1 < p < ∞ by operators of shift and multiplication [55,§ 1.1]:
g=(t,z): f(y) → [σ(t,z)f](y)=ei(2t−√2vy+uv) f(y− √2u),        z=u+iv,
(6.47)
i.e., this is the Schrödinger type representation with parameter (h/2p) = 1. These operators are isometries in Lp(Rn) and the adjoint representation ρ*(t,z)=ρ(−t,−z) in Lq(Rn), p−1+q−1=1 is given by a formula similar to (3.3).

6.3.2  Wavelet Transforms for the Heisenberg Group in Function Spaces

Example 1 We start from the subgroup G0=Rn+1={(t,z)  |   ℑ(z)=0}. Then X=G/G0=Rn and an invariant measure coincides with the Lebesgue measure. Mappings s: Rn → Hn and r: Hn → H are defined by the identities s(x)=(0,ix), s−1(t,z)=ℑz, r(t,u+iv)=(t,u). The composition law s−1((t,z)· s(x))=x+u reduces to Euclidean shifts on Rn. We also find s−1((s(x1))−1·s(x2))=x2−x1 and r((s(x1))−1·s(x2)) = 0.
We consider the representation σ(g) of Hn in the space of smooth rapidly decreasing functions B=S(Rn). As a character of G0=Rn+1 we take the χ(t,u)=e2it. The corresponding test functional l0 satisfying to 1.20iii is the integration l0(f)=(2π)−n/2∫Rn f(y) dy. Thus the wavelet transform is as follows
^
f
 
 (x)=⌠
⌡

Rn 
 σ(s(x)−1) f(y) dy = (2π)−n/2⌠
⌡

Rn 
 ei √2xy f(y) dy
(6.48)
and is nothing else but the Fourier transform3.
Now we arrive to the absence of a vacuum vector in B, indeed there is no a f(x) ∈ S(Rn) such that
[σ(t,u) f](y) = χ(t,u)f(y) ⇔ ei2t f(y− √2u)=ei2t f(y).
There is a way out accordingly to Subsection 1.3. We take B′=L∞(Rn) ⊃ B and the vacuum vector b0(y) ≡ (2π)−n/2 ∈ B′. Then coherent states are bx(y)=(2π)−n/2 e−i √2xy and the inverse wavelet transform is defined by the inverse Fourier transform
f(y) = ⌠
⌡

Rn 
^
f
 
 (y) bx(y) dx = (2π)−n/2 ⌠
⌡

Rn 
^
f
 
 (y) e−i √2xy dx.
The condition 1.20iv MW: B → B follows from the composition of two facts W: B→ B and almost identical to it M: B→ B, which are proved in standard analysis textbooks (see for example [32,§ IV.2.3]). To check scaling (1.14) according to the tradition in analysis  [26] we take a probe vector p0=e−y2/2 ∈ B. Due to well known formula ∫−∞+∞e−y2/2dy=(2π)1/2 of real analysis we have
 
  		⌠
⌡

X 
  
 
^
p
 
0 
 (x),l0  
  		bx dx,l0  
 
=
(2π)−n ⌠
⌡ 		⌠
⌡ 		⌠
⌡ 		ei√2xy e−y2/2 dy e−i√2xw dx dw
=
(2π)n/2 ⌠
⌡

Rn 
 e−y2/2dy
=
〈 p0,l0 〉.
Thus our scaling is correct. W and M intertwine the left regular representation - multiplication by ei √2yv with operators
[λ(g) f] (x)
=
χ(r(g−1·x)) f(g−1·x)
=
ei √2·0 f(x−√2u)=f(x−√2u),
i.e. with Euclidean shifts. From the identity 〈 W v , M* l 〉 F(X) = 〈 v,l 〉B (1.8) follows the Plancherel's identity:
⌠
⌡

Rn 
^
v
 
 (y)
^
l
 
 (y) dy
=
⌠
⌡

Rn 
 v(x) l(x) dx .
These are basic and important properties of the Fourier transform.
The Schrödinger representation is irreducible on S(Rn) thus M=W−1. Thereafter integral formulas (1.15) and (1.16) representing operators MW=WM=1 correspondingly give an integral resolution for a convolution with the Dirac delta δ(x). We have integral resolution for the Dirac delta
δ(x−y)=(2π)−n/2⌠
⌡

Rn 
 eiξ(x−y) dξ.
All described results on the Fourier transform are a part of any graduate curriculum. What is a reason for a reinvention of a bicycle here? First, the same path works with minor modifications for a function theory in R1,1 described in [34]. Second we will use this interpretation of the Fourier transform in Example 3.5 for a demonstration how the Weyl functional calculus fits in the scheme outlined in [33,35] and Subsection 2.2.
Remark 2 Of course, the Heisenberg group is not the only possible source for the Fourier transform. We could consider the "ax+b" group [55,Chap. 7] of the affine transformations of Euclidean space Rn. The normal subgroup G0=R of dilations generates the homogeneous space X=Rn on which shifts act simply transitively. The Fourier transform deduced from this setting will naturally exhibit scaling properties. We could alternatively consider a group Mn of Möbius transformation [13,Chap. 2] in Rn+1 which map upper half plane to itself. Then there is an induced action of Mn on Rn-the boundary of upper half plane. Mn generated by composition of the affine transformations and the Kelvin inverse [13,Chap. 2]. If we take the normal subgroup G0 generated by dilations and the Kelvin inverse then the quotient space X will again coincide with Rn and we immediately arrive to the above case. On the other hand the Fourier transform derived in such a way could be easily connected with the plane wave decomposition [51] in Clifford analysis [12,20].
Example 3 As a subgroup G0 we select now the center of Hn consisting of elements (t,0). Of course X=G/G0 isomorphic to Cn and mapping s: Cn → G simply is defined as s(z)=(0,z). The Haar measure on Hn coincides with the standard Lebesgue measure on R2n+1 [55,§ 1.1] thus the invariant measure on X also coincides with the Lebesgue measure on Cn. Note also that composition law s−1(g· s(z)) reduces to Euclidean shifts on Cn. We also find s−1((s(z1))−1·s(z2))=z2−z1 and r((s(z1))−1·s(z2)) = [1/2] ℑz1z2.
As a "vacuum vector" we will select the original vacuum vector of quantum mechanics-the Gauss function f0(x)=e−x2/2 which belongs to all Lp(Rn). Its transformations are defined as follow:
fg(x)=[ρ(t,z) f0](x)
=
ei(2t−√2vx+uv) e−(x− √2u)2/2
=
e2it−(u2+v2)/2 e− ((u−iv)2+x2)/2+√2(u−iv)x
=
e2it−zz/2e− (z2+x2)/2+√2zx.
Particularly [ρ(t,0) f0](x)=e−2itf0(x), i.e., it really is a vacuum vector in the sense of our definition with respect to G0. For the same reasons we could take l0(x)=e−x2/2 ∈ Lq(Rn), p−1+q−1=1 as the test functional.
It could be shown that [ρ(0,z) f0](x) belongs to Lq(Rn)⊗Lp(Cn) for all p > 1 and q > 1, p−1+q−1=1. Thus transformation (1.4) with the kernel [ρ(0,z)f0](x) is an embedding Lp(Rn) →Lp(Cn) and is given by the formula
^
f
 
 (z)
=
〈 f,ρs(z)f0 〉
=
π−n/4⌠
⌡

Rn 
 f(x) e−zz/2 e−(z2+x2)/2+√2zx dx
=
e−zz/2π−n/4⌠
⌡

Rn 
 f(x) e−(z2+x2)/2+√2zx dx .
(6.49)
Then ∧f(g) belongs to Lp( Cn , dg) or its preferably to say that function \brevef(z)=ezz/2∧f(t0,z) belongs to space Lp( Cn , e− | z |2 dg) because \brevef(z) is analytic in z. Such functions for p=2 form the Segal-Bargmann space F2( Cn, e− | z |2 dg) of functions [6,46], which are analytic by z and square-integrable with respect the Gaussian measure e− | z |2dz. For this reason we call the image of the transformation (3.5) by Segal-Bargmann type space Fp( Cn, e− | z |2 dg). Analyticity of \brevef(z) is equivalent to condition ( [ ∂/( ∂zj )] + [1/2] zj I )∧f(z)=0 .
The integral in (3.5) is the well-known Segal-Bargmann transform [6,46]. Inverse to it is given by a realization of (1.6):
f(x)
=
⌠
⌡

Cn  
^
f
 
 (z) fs(z)(x) dz
=
⌠
⌡

Cn  
^
f
 
 (u,v) eiv(u−√2x) e−(x−√2u)2/2  du dv
(6.50)
=
⌠
⌡

Cn  
 \brevef(z) e− (z2+x2)/2+√2zx e−| z |2 dz.
The corresponding operator P (1.7) is an identity operator Lp(Rn) →Lp(Rn) and (1.7) gives an integral presentation of the Dirac delta.
Integral transformations (3.5) and (3.6) intertwines the Schrödinger representation (3.3) with the following realization of representation (1.5):
λ(t,z) f(w)
=
^
f
 
0 
 (z−1·w)
-
χ
 
 (t+r(z−1·w))
(6.51)
=
^
f
 
0 
 (w−z)eit+iℑ(zw)
(6.52)
Meanwhile the orthoprojection L2( Cn, e− | z |2 dg) → F2( Cn, e− | z |2 dg) is of a separate interest and is a principal ingredient in Berezin quantization [9,16]. We could easy find its kernel from (1.10). Indeed, ∧f0(z)=e − | z |2 , then the kernel is
K(z,w)
=
^
f
 
0 
 (z−1·w)
-
χ
 
 (r(z−1·w))
=
^
f
 
0 
 (w−z)eiℑ(zw)
=
exp
 1
2
 (− | w−z |2 +w
-
z
 
 −z
-
w
 
 )
=
exp
 1
2
 (− | z |2− | w |2) +w
-
z
 

 .
To receive the reproducing kernel for functions \brevef(z)=e| z |2 ∧f(z) in the Segal-Bargmann space we should multiply K(z,w) by e(−| z |2+ | w |2)/2 which gives the standard reproducing kernel = exp(− | z |2 +wz) [6,(1.10)].

6.3.3  Operator Valued Representations of the Heisenberg Group

We proceed now with our main targets: wavelets in operator algebras. We shell show that well-known and new functional calculi are realizations of the scheme from Subsection 2.2.
Convention 4 [3] Let B be a Banach space. We will say that an operator A: B→ B is unitary if A is invertible and ||Ab||=||b|| for all b ∈ B. An operator A: B→ B is called self-adjoint if the operator exp(iA) is unitary. In the Hilbert space case this convention coincides with the standard definition.
Let T1, ..., Tn be an n-tuples of selfadjoint linear operators on a Banach space B. We put for our convenience T0=I-the identical operator. It follows from the Trotter-Daletskii4 formula [45,Thm. VIII.31] that any linear combination ∑j=0n aj Tj is again a selfadjoint operator. We will consider a set of unitary operators
T(a0,a1,…,an)=exp
 i n
∑
j=0 
 aj Tj
(6.53)
parametrized by vectors (a0,a1,…,an) ∈ Rn+1. Particularly T(0,0,…,0)=I. A family of their transformations ω(t,z), t ∈ R, z ∈ Cn is defined by the rule
ω(t,z)T(a0,a1,…,an) = T
 a0+t + n
∑
j=1 
 (uj vj −√2 aj uj ),
       a1− √2 v1, …, an− √2vn ),
(6.54)
where zj=uj+ivj. A direct calculation shows that ω(t′,z′)ω(t",z")=ω(t′+t"+[1/2]ℑ(z′z"),z′+z")-this is a non-linear geometric representation of the Heisenberg group Hn. We could observe that
T(a0,a1,…,an)=ω(a0, a) T(0,0,…,0) = ω(a0, a) T0 = ω(a0, a)I,
where a=(ia1,…,ian). Obviously all transformations ω(t,z) are isometries if the norm of elements T(a0,a1,…,an) is defined as their operator norm.
The representation ω (3.10) is not linear and we would like to use the procedure outlined in Remark 1.2. We construct the linear space of operator valued functions L(Rn,\mathfrakB) for a Hn-homogeneous space X as follows
[Tf](t)=⌠
⌡

X 
 f(x) ω(s(x)) dx  T(t),        t ∈ Rn, s(x) ∈ Hn.
(6.55)
We also extend the representation ω to L(X,\mathfrakB) as follows:
ω: [Tf](t) → ω(g) [Tf](t) = ⌠
⌡

X 
 f(g·x) ω(s(x)) dx  T(t),
(6.56)
where [Tf](t) ∈ L1(Hn), g ∈ Hn.
We will go on with coherent states defined by such a representation. In the notations of Subsection 2.2 operators T1, ..., Tn form a set T defining the representation τ = ω in (3.10) with T0(x)=I being a vacuum vector.
Example 5 We are ready to demonstrate that the Weyl functional calculus is an application of Definition 2.9 and Example 3.1 as was announced in [33,Remark 4.3]. Consider again the subgroup G0=Rn+1={(t,z)  |  ℑ(z)=0} and a realization of scheme from Subsection 1.3 for this subgroup. Then the first paragraph of Example 3.1 is applicable here.
We could easily see that ω(t,u1,…,un)I=eit+i∑1n ujI and we select the identity operator I times (2π)−n/2 as the vacuum vector T0(x) of the representation ω. Thereafter the transformation T: S(Rn) → L(Rn,\mathfrakB) (3.11) is exactly the inverse wavelet transform for the representation ω. This transformation is defined at least for all f ∈ S(Rn). The space S(Rn) is the image of the wavelet (Fourier) transform (3.4). Thus as outlined in Proposition 2.11 we could construct an intertwining operator F between σ (3.3) and ω (3.10) from the formula (2.20) as follow (see Remark 2.10):
[ΦT f] (0)
=
Mω Wσ f = (2π)−n/2 ⌠
⌡

Rn 
^
f
 
 (x) ω(0, x1,…, xn)  I dx
=
(2π)−n/2 ⌠
⌡

Rn 
^
f
 
 (x) ei∑1n xjTj  dx.
(6.57)
This formula is exactly the integral formula for the Weyl functional calculus [3,44,53]. As an example one could define a function
e−∑1n Tj2/2 = (2π)−n/2 ⌠
⌡

Rn 
 e−∑1n xj2/2 ei∑1n xj Tj  dx.
(6.58)
As we have seen (2.19) one could formally write the integral kernel from (3.13) as convolution of the integral kernels for Wσ and Mω:
Φ(y,0) = ⌠
⌡

Rn 
 e−i∑1n yj xj ei∑1n xj Tj dx.
(6.59)
This expression looks very formal, but it is possible to give it a precise mathematical meaning as an operator valued distribution. Such an approach was explored by ANDERSON in [3]. The support of this distribution was defined as the Weyl joint spectrum for n-tuple of non-commuting operators T1, ...,Tn and studied in [3].
Remark 6 As was mentioned in Remark 3.2 one could construct the Fourier transform from representations of ax+b group or the group Mn of Möbius transformations of the upper half plane. Analogously one could deduce the Weyl functional calculus as an intertwining operator between two representation of this group. The Cauchy kernel G(x) [12,§ 9] in Clifford analysis is the kernel of the Cauchy integral transform
f(y) = ⌠
⌡

∂X 
 Gy (x)
→
n
 
 (x) f(x) dσ(x),        y ∈ X,
where →n(x) is the outer unit vector orthogonal to ∂X and dσ(x) is the surface element at a point x. The Cauchy integral formula (as any wavelet transform) intertwines two representations acting on X and ∂X of Mn similarly to the case of complex analysis [35,§ 6]. Thus we could apply here Theorem 2.13 on a mapping of spectral distributions. The formula (2.28) take the form
ΦW f = ⌠
⌡

∂X 
 ΦW(G) (x)
→
n
 
 (x) f(x) dσ(x),
where ΦW stands for the Weyl functional calculus. This gives another interpretation for the main result of the paper [29,Thm. 5.4].
There is no a reason to restrict ourselves only to the case of subgroup G0=Rn+1={(t,z)  |  ℑ(z)=0}. Thus we proceed with the next example.
Example 7 In an analogy with Example 3.3 let us consider now the wavelet theory associated to the subgroup G0=R1={(t,0)} and the representation ω. The first paragraph of Example 3.3 depends only on G=Hn and G0=R1 and thus is applicable in our case.
It is easy to see from formula (3.10) that any operator valued function [Tf](x) (3.11) is an eigen vector for ω(h), h ∈ G0. To be concise with function models we select as a vacuum vector the operator exp(−∑1n T2j) (3.14). Then the condition (2.13) immediately follows from (3.14). Thus we could define a functional calculus Φ: Fp(Cn)→ L(Cn,\mathfrakB) by the formula (see Remark 2.10):
[ΦTf](0)
=
⌠
⌡

Cn 
 f(z) ω(0,z) exp(− n
∑
1 
 T2j/2) dz
=
⌠
⌡

Cn 
 f(z) ω(0,z) (2π)−[n/2]⌠
⌡

Rn 
 e−∑1n xj2/2ei∑1n xj Tj  dx dz
=
(2π)−[n/2] ⌠
⌡

Cn 
 f(z) ⌠
⌡

Rn 
 e−∑1n xj2/2 ω(0,z) ei∑1n xj Tj  dx dz
=
(2π)−[n/2] ⌠
⌡

R3n 
 exp n
∑
j=1 
− xj2
2
 +i(vj−√2xj) uj +i (xj − √2 vj)Tj
              ×f(z) dx du  dv.
(6.60)
The last formula could be rewritten for mutually commuting operators Tj as follows:
[ΦTf] (0)
=
(2π)−[n/2] ⌠
⌡

R2n 
 ⌠
⌡

Rn 
 exp n
∑
j=1 
− xj2
2
 + ixj (Tj−√2uj)
  dx
              ×exp n
∑
j=1 
 i(vjuj− √2 vjTj) f(z)  du  dv
=
(2π)−[n/2] ⌠
⌡

Cn 
 exp n
∑
j=1 
ivj(uj−√2Tj) − (Tj−√2uj)2
2

 f(z) dz
where the exponent of operator is defined in the standard sense, e.g. via the Weyl functional calculus (3.13) or the Taylor expansion. The last formula is similar to (3.6). This is very natural for commuting operators as well as that for non-commuting operators fromula (3.16) is more complicated.
The spectral distribution
ΦT(z,0)=(2π)−[n/2] ⌠
⌡

Rn 
 exp n
∑
j=1 
− xj2
2
 +i(vj−√2xj) uj +i (xj − √2 vj) Tj
  dx
derived from (3.16) contains at least as much information on operators T1, ..., Tn as the Weyl distribution (3.15) and deserves a careful separate investigation. We will just mention in conclusion that the Segal-Bargmann space is an example of the Fock space-space of second quantization for bosonic fields. Thus the functional calculus based on the Segal-Bargmann model sketched here seems to be an appropriate model for quantized bosonic fields.

Acknowledgements

I am grateful to Mohamed Bugatma for many suggestions, which help to improve these notes.

Bibliography

[1]
S. Twareque Ali, J.-P. Antoine, J.-P. Gazeau, and U.A. Mueller. Coherent states and their generalizations: A mathematical overview. Rev. Math. Phys., 7(7):1013-1104, 1995. Zbl # 837.43014.
[2]
Syed Twareque Ali, Jean-Pierre Antoine, and Jean-Pierre Gazeau. Coherent states, wavelets and their generalizations. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 2000.
[3]
Robert F. V. Anderson. The Weyl functional calculus. J. Functional Analysis, 4:240-267, 1969.
[4]
William Arveson. The Harmonic Analysis of Automorphisms Groups. American Mathematical Society, Amer. Math. Soc. Summer Institute, 1980.
[5]
Michael Atiyah and Wilfried Schmid. A geometric construction of the discrete series for semisimple Lie group. In J.A. Wolf, M. Cahen, and M. De Wilde, editors, Harmonic Analysis and Representations of Semisimple Lie Group, volume 5 of Mathematical Physics and Applied Mathematics, pages 317-383. D. Reidel Publishing Company, Dordrecht, Holland, 1980.
[6]
V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform. Part I. Comm. Pure Appl. Math., 3:215-228, 1961.
[7]
F. A. Berezin. Covariant and contravariant symbols of operators. Izv. Akad. Nauk SSSR Ser. Mat., 36:1134-1167, 1972. Reprinted in [9,pp. 228-261].
[8]
F. A. Berezin. Quantization. Izv. Akad. Nauk SSSR Ser. Mat., 38:1116-1175, 1974.
[9]
F. A. Berezin. Metod vtorichnogo kvantovaniya. "Nauka", Moscow, second edition, 1986. Edited and with a preface by M. K. Polivanov.
[10]
Felix A. Berezin. Quantization in complex symmetric spaces. Math. USSR-Izv., 9:341-379, 1975.
[11]
David Bernier and Keith F. Taylor. Wavelets from square-integrable representations. SIAM J. Math. Anal., 27(2):594-608, 1996. MR # 97h:22004.
[12]
F. Brackx, Richard Delanghe, and F. Sommen. Clifford Analysis, volume 76 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1982.
[13]
Jan Cnops. Hurwitz Pairs and Applications of Möbius Transformations. Habilitation dissertation, Universiteit Gent, Faculteit van de Wetenschappen, 1994. See also [14].
[14]
Jan Cnops. An introduction to Dirac operators on manifolds, volume 24 of Progress in Mathematical Physics. Birkhäuser Boston Inc., Boston, MA, 2002.
[15]
Jan Cnops and Vladimir V. Kisil. Monogenic functions and representations of nilpotent Lie groups in quantum mechanics. Mathematical Methods in the Applied Sciences, 22(4):353-373, 1998. E-print: arXiv:math/9806150. MR # 2000b:81044. Zbl # 1005.22003.
[16]
Lewis A. Coburn. Berezin-Toeplitz quantization. In Algebraic Mettods in Operator Theory, pages 101-108. Birkhäuser Verlag, New York, 1994.
[17]
Alain Connes. Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. (4), 6:133-252, 1973.
[18]
H.O. Cordes. Pseudodifferential Operators - An Abstract Theory, volume 756 of Lect. Notes Math. Springer-Verlag, Berlin, 1979.
[19]
Yu.L. Daletski. On representation of solutions of operator equations in a form of functional integrals. Dokl. Akad. Nauk SSSR, 134(5):1013-1016, 1960.
[20]
R. Delanghe, F. Sommen, and V. Soucek. Clifford Algebra and Spinor-Valued Functions, volume 53 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1992. A function theory for the Dirac operator, Related REDUCE software by F. Brackx and D. Constales, With 1 IBM-PC floppy disk (3.5 inch).
[21]
Jacques Dixmier. Les C*-algebres et Leurs Representations. Gauthier-Villars, Paris, 1964.
[22]
Jacques Dixmier. C\sp*-algebras. North-Holland Publishing Co., Amsterdam, 1977. Translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15.
[23]
M. Duflo and Calvin C. Moore. On the regular representation of a nonunimodular locally compact group. J. Functional Analysis, 21(2):209-243, 1976.
[24]
V. A. Fock. Konfigurationsraum und zweite quantelung. Z. Phys. A, 75:622-647, 1932.
[25]
Gerald B. Folland. Harmonic analysis in phase space, volume 122 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989.
[26]
Roger Howe. On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (N.S.), 3(2):821-843, 1980.
[27]
Roger Howe. Quantum mechanics and partial differential equations. J. Funct. Anal., 38(2):188-254, 1980.
[28]
Roger Howe and Eng Chye Tan. Non-Abelian Harmonic Analysis: Applications of SL(2,R). Universitext. Springer-Verlag, New York, 1992.
[29]
Brian Jefferies and Alan McIntosh. The Weyl calculus and Clifford analysis. Bull. Austral. Math. Soc., 57(2):329-341, 1998.
[30]
L. V. Kantorovich and G. P. Akilov. Functional analysis. Pergamon Press, Oxford, second edition, 1982. Translated from the Russian by Howard L. Silcock.
[31]
A. A. Kirillov. Elements of the theory of representations. Springer-Verlag, Berlin, 1976. Translated from the Russian by Edwin Hewitt, Grundlehren der Mathematischen Wissenschaften, Band 220.
[32]
Alexander A. Kirillov and Alexei D. Gvishiani. Theorems and Problems in Functional Analysis. Problem Books in Mathematics. Springer-Verlag, New York, 1982.
[33]
Vladimir V. Kisil. Möbius transformations and monogenic functional calculus. Electron. Res. Announc. Amer. Math. Soc., 2(1):26-33, 1996. (electronic) MR # 98a:47018.
[34]
Vladimir V. Kisil. Analysis in R1,1 or the principal function theory. Complex Variables Theory Appl., 40(2):93-118, 1999. E-print: arXiv:funct-an/9712003. MR # 2000k:30078.
[35]
Vladimir V. Kisil. Two approaches to non-commutative geometry. In H. Begehr, O. Celebi, and W. Tutschke, editors, Complex Methods for Partial Differential Equations, chapter 14, pages 219-248. Kluwer Academic Publishers, Netherlands, 1999. E-print: arXiv:funct-an/9703001, MR # 2001a:01002.
[36]
A.W. Knapp and N.R. Wallach. Szegö kernels associated with discrete series. Invent. Math., 34(2):163-200, 1976.
[37]
M. G. Kren and M. A. Rutman. Linear operators leaving invariant a cone in a Banach space. Uspehi Matem. Nauk (N. S.), 3(1(23)):3-95, 1948. MR # 10:256c.
[38]
Serge Lang. Algebra. Addison-Wesley, New York, 1969.
[39]
Serge Lang. SL\sb 2(R), volume 105 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1985. Reprint of the 1975 edition.
[40]
George W. Mackey. Mathematical Foundations of Quantum Mechanics. W.A. Benjamin, Inc., New York, Amsterdam, 1963.
[41]
Stéphane Mallat. A wavelet tour of signal processing. Academic Press Inc., San Diego, CA, 1998. Illustration from the book are on-line: http://www.cs.nyu.edu/cs/faculty/mallat/book.html.
[42]
Willard Miller, Jr. Lie Theory and Special Functions. Academic Press, New York, 1968. Mathematics in Science and Engineering, Vol. 43.
[43]
Vladimir Nazaikinskii and Boris Sternin. Wave packet transform in symplectic geometry and asymptotic quantization. In Komrakov B.P., Krasil'shchik I.S., Litvinov G.L., and Sossinsky A.B., editors, Lie Groups and Lie Algebras. Their Representations, Generalizations and Applications, number 433 in Mathematics and Its Applications, pages 47-70, Dordrecht-Boston-London, 1998. Kluwer Academic Publishers.
[44]
E. Nelson. A functional calculus for non-commuting operators. In F.E. Browder, editor, Functional analysis and related fields, Proceedings of a conference in honour of Professor Marshal Stone, Univ. of Chicago, May 1968, pages 172-187. Springer-Verlag, Berlin, Heidelberg, New York, 1970.
[45]
Michael Reed and Barry Simon. Functional Analysis, volume 1 of Methods of Modern Mathematical Physics. Academic Press, Orlando, second edition, 1980.
[46]
Irving E. Segal. Mathematical Problems of Relativistic Physics, volume II of Proceedings of the Summer Seminar (Boulder, Colorado, 1960). American Mathematical Society, Providence, R.I., 1963.
[47]
Irving E. Segal. Algebraic quantization and stability. In William B. Arveson and Ronald G. Douglas, editors, Operator Theory: Operator Algebras and Applications, number 51 in Proceedings of Symposia in Mathematics, pages 503-518. American Mathematical Society, Providence, R.I., 1990.
[48]
Irving E. Segal. C*-algebras and quantization. In Robert Doran, editor, C*-Algebras: 1943-1993, number 167 in Contemporary Mathematics, pages 55-65. American Mathematical Society, Providence, R.I., 1994.
[49]
Irving E. Segal. Rigorous covariant form of the correspondence principle. In William Arveson et al., editors, Quantization, Nonlinear Partial Differential Equations, and Operator Algebra (Cambridge, MA, 1994), volume 59 of Proc. Sympos. Pure Math., pages 175-202. American Mathematical Society, Providence, R.I., 1996.
[50]
M. A. Shubin. Pseudodifferential operators and spectral theory. Springer-Verlag, Berlin, second edition, 2001. Translated from the 1978 Russian original by Stig I. Andersson.
[51]
Frank Sommen. Plane wave decompositions of monogenic functions. Annales Pol. Math., 49:101-114, 1988.
[52]
Masamichi Takesaki. Structure of Factors and Automorphism Groups, volume 51 of Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1983.
[53]
Michael E. Taylor. Functions of several self-adjoint operators. J. Funct. Anal., 19:91-98, 1968.
[54]
Michael E. Taylor. Pseudodifferential operators, volume 34 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1981.
[55]
Michael E. Taylor. Noncommutative harmonic analysis, volume 22 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1986.
[56]
N. Ja. Vilenkin. Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, R. I., 1968. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22.

Index (showing section)

SB2(C), 1.3
Λ(g), 3.2
SL2(R), 2.1
Rn±, 2.2
δjk, 3.1
〈 ·,· 〉, 3.1
tr , 3.1
identity
     Pythagoras
, 1.1

abstract group, 2.1
action
     transitive, 2.2
adjoint representation, 3.1
annihilation operator, 5.1
associativity, 2.1

Bergman space, 1.2

Cauchy integral formula, 1.2, 4.1
character
     of group, 3.1
character of a group, 3.1
character of representation, 3.1
coherent states, 0.0, 1.6, 4.1
commutative, 2.1
commuting operator, 3.3
conjugation, 2.2
continuous group, 2.1
convolution, 2.2
creation operator, 5.1, 5.3
cyclic vector, 3.2
dicrete series, 4.2
Dirac
     bra and ket, 1.1
dual object, 3.2
dual space, 3.2

equation Schrödinger, 5.3

finite dimensional representation, 3.1

generating function, 5.3
ground state, 4.1
group, 2.1
     SL2(R), 2.1
     ax+b, 2.1
     abstract, 2.1
     commutative, 2.1
     continuous, 2.1
     Heisenberg, 2.1
     Lie, 2.1
     noncommutative, 2.1
     of characters, 3.1
     representation, 3.1
     subgroup
         isotropy, 2.2
     transformation, 2.1
group multiplication, 2.1
Haar measure, 2.2
Hardy space, 1.2
Heisenberg group, 2.1
     Schrödinger representation, 3.1
Hilbert space, 1.1
homogeneous space, 2.2

identity, 2.1
     Parseval(-Pythagoras), 1.1
     Plancherel, 1.1
infinite dimensional representation, 3.1
invariant subspace, 3.2
inverse, 2.1
inverse wavelet transform, 4.1
irreducible representation, 3.2

kernel
     reproducing, 4.2
Kroneker delta, 3.1

Lebesgue measure, 2.2
left shift, 2.2
lemma
     Schur's, 3.3
locally compact, 2.1
loop, 2.2
lower (upper) half plane, 2.2
matrix elements, 3.1
measure, 2.2
     Haar, 2.2
     Lebesgue, 2.2
     left invariant, 2.2
measurement, 5.1
momentum space, 5.1
mother wavelet, 1.6

noncommutative, 2.1
nontrivial invariant subspaces, 3.2

observables, 5.1
operator
     annihilation, 5.1, 5.3
     commuting, 3.3
     convolution, 2.2
     creation, 5.1, 5.3
     intertwining, 3.3
     Töplitz, 1.2
     unitary, 3.1
orbits, 2.2
orthogonal polynomials, 5.3
orthonormal base, 1.1
orthonormal basis, 3.1
projection
     Bergman, 1.2
     Segal-Bargmann, 1.3

relations
     commutation, 5.1
representaion
     linearization, 3.1
representation, 3.1
     adjoint, 3.1
     continuous, 3.1
     exact, 3.1
     faithful, 3.1
     finite dimensional, 3.1
     infinite dimensional, 3.1
     irreducible, 3.2
     linear, 3.1
     reducible, 3.2
     regular, 3.2
     Schrödinger, 3.1
     series
         discrete, 4.2
     square integrable, 4.0, 4.2
     trivial, 3.1
     unitary, 3.1
representation of a group, 3.1
representation space, 3.1
representations, 3.1
     equivalent, 3.1
         unitary, 3.1
reproducing kernel, 1.2, 4.2
restriction of representation, 3.2
Riesz-Dunford functional calculus, 1.5
right shift, 2.2
scalar product, 3.1
Schrödinger representation, 5.1
Schur's lemma, 3.3
set
     total, 4.1
shift
     left, 2.2
     right, 2.2
singular integral operator
     Szegö, 1.2
space
     Bergman, 1.2
     Hardy, 1.2
     Hilbert, 1.1
     Segal-Bargmann, 1.3
square integrable modulo subgroup, 5.0
state
     ground, 4.1
states, 5.1
Stone-von Neumann theorem, 5.1
subrepresentation, 3.2
subspace
     invariant, 3.2
Töplitz operator, 1.2, 1.3
theorem
     Stone-von Neumann, 5.1
trace, 3.1
transform
     wavelet, 4.1
transformation
     linear-fractional, 2.2
transformation group, 2.1
transitive, 2.2
trivial representation, 3.1

unit circle, 1.2
unit disk, 1.2
unitary operator, 3.1
unitary representation, 3.1
upper (lower) half plane, 2.2

vacuum vector, 1.6, 4.1
vector
     admissible, 4.2
     cyclic, 3.2
     fiducial, 4.1
     vacuum, 1.6, 4.1
wave functions, 5.1
wavelet
     mother, 1.6, 4.1
wavelet transform, 4.1
     inverse, 4.1
wavelets, 0.0, 1.0, 1.6, 4.1

Footnotes:

1Nature is horrified by (any) vacuum (Lat.).
2Nature is horrified by a carrier of nothingness (Lat.). This illustrates how far a humane beings deviated from Nature.
3The inverse Fourier transform in fact. In our case the signs selection is opposite to the standard one, but we will neglect this difference.
4The formula is usually attributed to Trotter alone. It is widely unknown that the result appeared in [19] also.

File translated from TEX by TTH, version 3.67.
On 31 May 2006, 15:50.