Syllabus

Introduction to Hilbert spaces, the spaces I²(Z) and L²([-π,π]) , Parseval’s identity, pointwise convergence of the Fourier series-criteria, principle of localization,L^∞-estimates of the partial sums of the Fourier series, the Schwartz space S (R), the Fourier transform on  S (R), properties of the Fourier transform, the inversion theorem, convolution and approximations of the identity on  R, the Fourier transform as an operator on  S (R), Plancherel theorem, the Poisson summation formula, the Heisenberg uncertainty principle.

Course Outline