Topics in Geometry or Topology

(3 Credit Hours) · Undergraduate

Fourier Analysis is one of the fundamental topics of modern mathematics, whose origin lies in the work of Euler and the Bernoulli brothers and their study of the vibrating string. It has, since, become an essential tool in mathematics, physics, engineering, computer science, etc, with applications from PDE to number theory to representation theory to combinatorics. This course will begin with the classical theory of Fourier series followed by an overview of Fourier transforms in one and several variables, with an emphasis on applications. Possible topics include: convolutions and summability kernels - Fejer, Poisson; convergence of Fourier series - Hilbert spaces and mean-square and pointwise convergence; applications; Fourier transform on the line: the Schwartz space; Fourier inversion formula; Plancherel theorem; Poisson summation formula; applications; Fourier transform on R^n.