### MA6080 Fourier Analysis (modified)

#### Course Details

Unit 1: The genesis of Fourier Analysis, Fourier Series, Uniqueness of Fourier coefficients, Pointwise and Uniform convergence of Fourier series, Convolution in L^{1}(T), Cesaro and Abel summability of Fourier series, Fourier series in L^{2}(T), A continuous function with a divergent Fourier series, Applications to Weierstrass Approximation theorem and The Isoperimetric Problem. (15 lectures)

Unit 2: Schwartz space on R, Fourier transform on the Schwartz space, Fourier transform on L^{2}(R); Fourier transform on L^{1}(R), Density of D(R) in L^{p}(R); 1 ≤ p < ∞. (12 lectures)

Unit 3: Poisson summation formula, Heisenberg uncertainty principle, Hardy's theorem, Paley-Wiener theorem, An introduction to tempered distributions, Differentiation, convolution and Fourier transform on the class of tempered distributions. (13 lectures)

#### Course References:

Text Books:

1.E. M. Stein and R. Shakarchi, Fourier Analysis - An Introduction, Princeton Lectures in Analysis, Volume 1, 2003.

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

Reference Books:

1.R. Radha and S. Thangavelu, Fourier Analysis, Lecture Notes, 2012,
https://nptel.ac.in/courses/111106046/

2.M.A. Pinsky, Introduction to Fourier Analysis and Wavelets, GTM vol 102, AMS, 2008.

3.E. M. Stein and G. Weiss, An introduction to Fourier Analysis on Euclidean spaces,
Princeton University Press, 1971.

4.Y. Katznelson, An introduction to Harmonic Analysis, Cambridge University Press,
3rd Edition, 2002.