Algebra 2

FOURIER ANALYSIS

Lecturer, TA
Eitan Sayag

Fourier Series of continuous periodic functions. Uniqueness principle and uniform Convergence for C^2.
Convolution, Summability Kernels, Fejer theorem. Cesaro and Abel convergence. Poisson Kernel.
Applications: Weyl's criterion for equidistribution and Benfold's Law.
Riemann-Lebesgue's Lemma, Dini's Criterion, Riemann's localization principle.
Hardy Tauberian Theorem, Jordan's version of Dirichlet's theorem.
Fourier Series of Riemann integrable functions. Discontinuities, Gibbs Phenomena.
History: Du Bois-Reymond's Example and Kolmogorov's Example. Carlson Theorem.
A Banach-Steinhaus look at the pointwise divergence of Fourier series.
Convergences in L2. Bessel inequality, Parseval's identity. Generalizations to separable Hilbert spaces (basis, completeness).
Applications: isoperimetric inequality, calculating values of zeta.
Applications: Solution of the heat equation, wave equation and Laplace equation. The age of the Earth. Hear This
Existence of trigonometric series representing wild continuous functions. The limitations of Fourier Expansions.
Periodic Distributions and Sobolev Spaces.
Fourier Transform
Fourier Transform on Schwartz space and on L2 of the line. Plancherel formula.
Distributions. Solving PDE via Fourier method.
Poisson summation formula and applications to geometry of numbers.
Applications to Communication systems. Signal Processing.
Multi-dimensional theory and its applications: Radon Transform.

The Syllabus is here.

I. Stein and R. Shakarchi, Introduction to Fourier Analysis: An introduction, (Princeton Lectures in Amnalysis), Princeton University Press, 2003.
T. W. Korner, Fourier Analysis, Cambridge University Press, 1989
T. W. Folland, Fourier Analysis and Its applications, Cambridge University Press, 1989