MTL743:Fourier Analysis

3 Credits (3-0-0)

Pre-requisites: MTL122/MTL503

Fourier Series - Definition, uniqueness, convolution, summability, convergence of Fourier series, Fourier series for square integrable functions, Plancheral theorem, Riesz-Fischer theorem, Gibb’s phenomenon, divergence of Fourier series Applications of Fourier series – Isoperimetric inequality, Weierstrass approximation theorem, Weyl’s equidistribution theorem, heat equation on the circle. Fourier transform – Schwartz space on R, Fourier transform on the Schwartz space, Fourier transform of integrable and square-integrable functions, Poisson summation formula. Tempered distributions – Topology on the Schwartz space, tempered distributions and its properties, Fourier transform of tempered distributions. Applications – Uncertainty principle, Paley-Wiener theorem, Wiener’s theorem, Shannon sampling theorem, multiplier theorem for integrable functions.