MA7013 Fourier Analysis on Eucidean Spaces

Course Details

To study in detail the theoretical aspects of Fourier transforms which is useful in various subjects of Science and Engineering

Fourier transform and tempered distributions: Review of the L1 and L2 theory of Fourier transform, Schwartz class of rapidly decreasing functions S(R), Topology on S(R), the class of tempered distributions, calculus of tempered distributions, convolution and Fourier transform of tempered distributions.
Some primary theorems in Fourier analysis: Qualitative and quantitative uncertainty principle, Hardy’s theorem, Paley-Wiener theorem, Wiener’s theorem, Wiener-Tauberian theorem, spherical harmonics.
Introduction to singular integrals: Maximal function, integral of Marcinkiewicz, decomposition in cubes of open sets in Rn, an interpolation theorem for Lp, Carlderon-Zygmund kernel, Lp boundedness of a convolution operator with singular kernel.

Course References:

1. Steven G. Krantz, A panorama of Harmonic Analysis, The Mathematical Association of America, 1999.
2. W. B. Rudin, Functional analysis, Tata McGraw Hill, 1991.
3. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1986.
4. H. Dym and H. P. McKean, Fourier series and integrals, Academic Press, 1972.

Prerequisite:Fourier Analysis ( MA 6080) or equivalent