**Postgraduate course in Analysis
Second semester, 2008-2009**

Wednesdays 1 p.m. to 2 p.m. in Roger Stevens LT10, and 2 p.m. to 3 p.m. in Social Studies 9.01.

**PROVISIONAL SYLLABUS**

1. Introduction. Examples of such spaces (Hardy spaces, Bergman spaces, Wiener algebra, Paley-Wiener space). (1)

2. Hardy spaces on the disc. Poisson kernel. Inner and outer functions. (5)

3. Operators on H^{2} and L^{2}. Laurent, Toeplitz and Hankel operators.
Nehari, Carathéodory--Fejér and Nevanlinna--Pick problems. Hilbert
transform. (5)

4. Hardy spaces on the half-plane. Laplace and Fourier transforms. Theorems of Paley--Wiener and Plancherel. (2)

5. Commutative Banach algebras. Gelfand theory, applied to the disc
algebra, Wiener algebra and H^{}. (5)

6. Reproducing kernel Hilbert spaces. Whittaker--Kotel'nikov--Shannon sampling theorem. More on interpolation as time permits. (4)

**BOOKS**

K. Hoffman, Banach spaces of analytic functions.

P. Koosis, Introduction to H_{p} spaces.

W. Rudin, Real and complex analysis.

N. Nikolski, Operators, functions and systems, an easy reading, Vol. 1.

**PRE-REQUISITES**

Familiarity with the main theorems of elementary complex analysis. Some experience of Hilbert spaces and the concept of a bounded linear operator. The definition, at least, of a Banach space.

Last updated August 31st 2009