Course Overview: Distribution theory can be thought of as the completion of differential calculus, just as Lebesgue integration theory can be thought of as the completion of integral calculus. It was created by Laurent Schwartz in the 20th century, as was Lebesgue's integration theory.
Distribution theory is a powerful tool that works very well in conjunction with the theory of Fourier transforms. One of the main areas of applications is to the theory of partial differential equations. In this course we give an introduction to these three theories.
Lecturer(s):
Prof. Jan Kristensen
Learning Outcomes: Students will become acquainted with the basic techniques that in many situations form the starting point for the modern treatment of PDEs.
Course Synopsis: Test functions and distributions on \(\mathbb{R}^n\): definitions and examples, Dirac \(\delta \)- function, approximate identities and constructions using convolution of functions.[3 lectures]
The calculus of distributions on \(\mathbb{R}^n\) : functions as distributions, operations on distributions, adjoint identities, consistency of derivatives, distributional and weak solutions of PDEs, Sobolev functions.[3 lectures]
The Fourier transform on \(\mathbb{R}^n\): from Fourier series to Fourier integrals (only for \( n = 1\)), the Schwartz class \(\mathcal{S}\) of test functions on \(\mathbb{R}^n\) , properties of the Fourier transform on \(\mathcal{S}\), the Fourier transform of a Gaussian and the inversion formula on \(\mathcal{S}\). [3 lectures]
Fourier transforms of tempered distributions: definitions and examples, convolutions with tempered distributions.[2 lectures]
Solving PDEs using Fourier transformation: the Laplace equation, the heat equation, the wave equation, Schrödinger's equation.[2 lectures]
Fourier Analysis: the Riemann-Lebesgue lemma, Paley-Wiener theorems, the Poisson summation formula, the uncertainty principle.[3 lectures]
