Syllabus:

Module 1.
Definition and examples. The problem of convergence of a Fourier series. Dirichlet kernel, Poisson Kernel. Fourier series of a twice continuously differentiable function. Convolutions, Good kernels.

Module 2.
Riemann-Lebesgue lemma. Convergence at the points of differentiability. Continuous

Module 3.
The Fourier transform on R and its properties. The Schwartz space. The Gaussian as a good

Module 4.
Weierstrass approximation theorem. Heat equation on the real line. Steady state heat function with diverging Fourier series. Isoperimetric inequality and some applications. kernel. Fourier inversion and Plancherel’s formula. equation in the upper half plane. Mean value property.

Abstract of the Lectures:

(i) V. M. Sholapurkar: Introduction to Fourier series, definition and examples. The problem of convergence of a Fourier series. Convergence of a Fourier series under smoothness conditions. Convolutions, definition and examples of good kernels. The lectures will cover Module 1.

(ii) Swagato Roy: Convergence of the Fourier series at the points of differentiability. Continuous function with diverging Fourier series. Riemann-Lebesgue lemma. Some applications including the Isoperimetric inequality. The lectures will cover Module 2.

(iii) Dr. Sameer Chavan: The notion of Fourier transform on R and its properties will be discussed. The restriction of Fourier transform on Schwartz space. Fourier inversion and Plancherel’s formula. The lectures will cover Module 3.

(iv) Parasar Mohanty : Weierstrass approximation theorem. Mean value property. Heat equation on the real line. Steady state heat equation in the upper half plane. The lectures will cover Module 4.

First Week: 14^th to 19^th November, 2016

Second Week: 21^st to 26^th November, 2016

SC: Sameer Chavan
SR: Swagato Roy
VS: Vinayak Sholapurkar
PM: Parasar Mohanty
KM: Krishnat Masalkar
MS: Makarand Sarnobat