250135 VO Theory of distributions (2019S)

The theory of distributions (generalized functions), is an extension of classical analysis that was developed in the mid 20th century mainly by Laurent Schwartz and Sergei Sobolev. While in Physics and Engineering generalised-function-ideas (in a more or less vague sense) had been around much longer (Kirchhoff 1882, Heaviside 1898, Dirac 1926) it was the elegant functional analytic formulation due to Schwartz (around 1945), as well as the connection to Fourier analysis, that turned distribution theory into an indispensable tool in applications such as the theory of partial differential equations, harmonic analysis and theoretical physics.

Particularly in PDE, distributions became widespread since in many cases it is easier to establish the existence of distributional solutions than of classical ones, or appropriate classical solutions simply may not exist. In particular, the notion of fundamental solution allows one to formalize the practical use of `Green functions' which, of course, had appeared much earlier.

Although the quest for a general solution concept for linear PDE was the main driving force behind the development of distribution theory, it had truly unexpected and revolutionary consequences for the analysis of (linear) PDE, mainly due to the work of Lars Hörmander since the mid 1950ies (e.g. microlocal analysis). Today the theory of distributions is still an indispensable tool in the theory of PDE and beyond, e.g. in time frequency analysis, and forms the foundation of several important branches of modern analysis.

This lecture course provides a general introduction to the theory of distributions, with a particular emphasis on the calculus of generalized functions and its application to the determination of fundamental solutions. We will mainly follow the book [OW].