Fourier Series: Definition, Examples, Uniqueness of Fourier series,
Convolution, Cesaro summability and Abel summability of Fourier series,
Mean square convergence of Fourier series, A continuous function with
divergent Fourier series.
Some applications of Fourier series: The isoperimetric inequality, Weyl's
equidistribution theorem.
Fourier transform: The Schwartz space, Fourier transform on the real
line and basic properties, Approximate identity using Gaussian kernel,
Solution of heat equation, Fourier inversion formula, L²- theory , The
class of test functions, Distributions, Convergence, 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's theorem, WienerTauberian theorem.
- Teacher: mtnair THAMBAN NAIR M