Course Details
Subject {L-T-P / C} : MA5110 : Fourier Analysis { 3-0-0 / 3}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
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.
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.
Some basic theorems of Fourier Analysis: Poisson summation formula, Heisenberg uncertainty principle, Hardy’s theorem, Paley-Wiener theorem, Wiener-Tauberian theorem.
Course Objectives
- 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.
Course Outcomes
Students will learn about various types of convergence of Fourier series, properties of fourier transforms of functions in L^2(R) and generalized functions, along with the basic results of Fourier theory.
Essential Reading
- H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press
- T. W. Körner, Fourier Analysis, Cambridge University Press
Supplementary Reading
- 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