Course Archives Theoretical Statistics and Mathematics Unit

Course: Fourier Analysis Level: Postgraduate Time: Currently not offered

Syllabus Past Exams Syllabus: i) Fourier Series on T: a) Dirichlet problem for the unit disc and origin of Fourier series, continuity of translation on Lp(T) and elementary convolution inequalities, approximate identity. b) Fourier series and its elementary properties, completeness of trigonometric polynomials and Riemann-Lebesgue lemma, Uniqueness of Fourier coefficients and the Fourier inversion, Plancherel theorem, Weyls equidistribution theorem. c) Dirichlet kernel and pointwise convergence of Fourier series for Lipschitz continuous functions, Riemanns localization principle, existence of a continuous function with divergent Fourier series. d) Cesaro and Abel summability, Poisson integral and solution of Dirichlet problem for appropriate function classes. e) Marcinkiewicz interpolation theorem, Youngs inequality and Hausdorff- Young inequality, norm convergence of Fourier series for Lp, 1 < p < 8. ii) Fourier transform on Rd: a) Elementary properties of the Fourier transform involving translation, dilation, rotation, decay and smoothness, Riemann Lebesgue lemma, Fourier transform of Gaussian and the Poisson kernel, the Fourier inversion. Schwartz class functions and its image under Fourier transform. b) Fourier transform of L2-functions and the Plancherel theorem, Hausdorff Young inequality, Paley-Wiener theorem, Poisson summation formula. c) Tempered distribution and its Fourier transform, computation of some dis- tributional Fourier transform. d) Weak Lp spaces, Method of maximal function, Lebesgue differentiation theorem, almost everywhere convergence of Poisson integrals. e) Nontangential convergence of Poisson integral and characterization of Poisson integral of Lp functions (If time permits) Suggested Texts : (a) Real and complex analysis- W. Rudin. (b) Topics in Functional analysis- S. Kesavan. (c) Introduction to Fourier Analysis on Euclidean spaces- E. M. Stein, G. Weiss. (d) Fourier Analysis- E. M. Stein, R. Shakarchi. (e) Introduction to Harmonic Analysis-Y. Katznelson Top of the page Past Exams Top of the page

[ Semester Schedule ][ SMU ] [Indian Statistical Institute]