Trigonometric Series, Generalized Periodic Functions, and Boundary Value Problems of Mathematical Physics

摘要

A number of concrete problems of mathematical physics have led to the study of trigonometric series. While in classical analysis the series whose coefficients tend to zero were considered, the Sobolev-Schwarts distribution theory made it possible to give a meaning to trigonometric series with coefficients that grow in power fashion. The theory of ultradistributions and hyperfunctions afforded the possibility of including into consideration the series with coefficients having a higher order of growth. We indentify an arbitrary trigonometric series with a certain generalized periodic function, and give a characterization of some subclasses of generalized functions in terms of the growth of the coefficients of their Fourier series. Classical problems of mathematical physics are considered: the Dirichlet problem for the Laplace equation in a disk and the periodic Cauchy problem for the heat equation. It is shown that they acquire a natural formulation precisely within the framework of formal trigonometric series.