Analysis IV (Fourier Theory and Hilbert Spaces)

Abstract

This class covers the basic theory of Hilbert spaces, Fourier series and Fourier Transform, and its application to the study of classical linear PDEs.

Objective

1) Learn the basic theory of Hilbert spaces, Fourier series, Fourier Transform. Understand the strong physical intuition behind these mathematical concepts. 2) Learn about some concrete problems that can be effectively attacked with these tools, and understand what is the rigorous interpretation of the abstract results in such problems. Get a feeling about how to recognize such problems. 3) Learn what are the typical limitations and shortcomings of these tools.

Content

1) Real and complex Hilbert spaces, Hilbert bases and Riesz representation Theorem 2) Fourier series of a function in L^2([-π, π]; C), relationship between the regularity of a function and the asymptotic behaviour of the Fourier coefficients. Application to the resolution of linear partial differential equations with various boundary conditions in [-π, π]. 3) Fourier Transform in R^d and its elementary properties, relationship between the regularity of the function and the asymptotic behaviour of its Fourier transform, relationship between the summability of the function and the regularity of it Fourier transform. Application to the resolution of linear partial differential equations with various decay conditions in R^d. 4) Compact operators on Hilbert spaces, Self-adjoint operators, the spectral theorem, eigenvalue problems, and applications.

Resources