Course description: The first part of the course deals with the motivation for Fourier series and with the convergence issues that arise, while highlighting some applications. The second part will focus on the Fourier transform and its applications to classical partial differential equations and the Radon transform.

Topics covered:

• Basics: the origin of Fourier series. Uniqueness. Convergence for nice functions.
• Convolutions and summability: Good kernels. Fejér's theorem. The Poisson kernel.
• Further questions: Mean square and pointwise convergence.
• Applications in pure mathematics: the isoperimetric inequality, equidistribution, theta-functions.
• The Fourier transform on R: The Schwartz space, Fourier inversion, Plancherel formula and applications.
• The Fourier transform in R^d and applications.