Module I (10 hours)
Fourier Series: Definition, Examples, Uniqueness of Fourier Series, Convolution, Cesaro and Abel summability of Fourier Series, Fejer’s theorem, Poisson kernel and Dirichlet problem in the unit disc, Mean square convergence of Fourier series.

Module II (10 hours)
Fourier Transform: The Schwartz space, Fourier transform on the real line and basic properties, Fourier inversion formula, L^(2) theory, The class of test functions, Distributions, differentiation and convolution of distributions, Tempered distributions, Fourier transform of a tempered distribution.

Module III (10 hours)
Some basic theorems of Fourier Analysis: Poisson summation formula, Heisenberg uncertainty principle, Hardy’s theorem, Paley-Wiener theorem, Wiener-Tauberian theorem.

To cover different types of convergence of Fourier series, and related results.

To study Fourier transform and it's properties for square integrable function on R.

To study convolution, fourier transform etc. of distributions.

CO1. Students will learn about various types of convergence of Fourier series.

CO2. They will study properties of Fourier transforms of functions in L^2(R) and generalized functions.

CO3. They will get acquainted with the basic results of Fourier theory with their applications.

CO4. They will get ready to take advance courses on Fourier analysis on groups/fields.

H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press

T. W. Körner, Fourier Analysis, Cambridge University Press

G. B. Folland, Fourier Analysis and Its Applications, Brooks/Cole Publishing Co.

E. M. Stein and R. Shakarchi, Fourier Analysis: An introduction, Princeton University Press