MA6080 Fourier Analysis (modified)
Unit 1: The genesis of Fourier Analysis, Fourier Series, Uniqueness of Fourier coefficients, Pointwise and Uniform convergence of Fourier series, Convolution in L1(T), Cesaro and Abel summability of Fourier series, Fourier series in L2(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 L2(R); Fourier transform on L1(R), Density of D(R) in Lp(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)
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.
1.R. Radha and S. Thangavelu, Fourier Analysis, Lecture Notes, 2012,
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.