Course Details

Fourier Series: Definition, Examples,Uniqueness of Fourier series, Convolution, Cesaro summability and Abel summability of Fourier series, Mean square convergence of Fourier series, A continuous function with divergent Fourier series.
Some applications of Fourier series: The isoperimetric inequality, Weyl's equidistribution theorem. Fourier transform: The Schwartz space Fourier transform on the real line and basic properties, Approximate identity using Gaussian kernel, Solution of heat equation, Fourier inversion formula, L2- theory , The class of test functions, Distributions, Convergence, differentiation andconvolution of distributions, Tempered distributions, Fourier transform of a tempered distribution.
Some basic theorems of Fourier Analysis:} Poisson summation formula, Heisenberg uncertainty principle, Hardy's theorem, Paley-Wiener theorem, Wiener's theorem, Wiener-Tauberian theorem.

Course References:

H. Dym and H. P. McKean, Fourier series and Integrals, Academic press, 1972