Spaces of analytic functions

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 H2 and L2. 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 Hinfinity. (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 Hp 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.

Course notes (PDF)

Problem Sheets 1 and 2 (PDF)


Jonathan Partington

Last updated August 31st 2009