The course will cover L1, L2, L°° and basic facts from Hilbert space theory (Hilbert basis, projection theorems, Riesz theory). The first part of the course will introduce Fourier series: the L2 theory, the C°° theory: rate of convergence, Fourier series of real analytic functions, application to the trapezoidal rule, Fourier transforms in L1, Fourier integrals of Gaussians, the Schwartz class S, Fourier transforms and derivatives, translations, convolution, Fourier transforms in L2, and characteristic functions of probability distribution functions. The second part of the course will cover tempered distributions and applications to partial differential equations. Other related topics will be covered at the instructor’s discretion.
Prerequisite Courses