A New Look at the Schwartz Theory

Abstract

The main function of the present chapter is to construct local analogues of the Schwartz theory of mean periodic functions. This theory is a study of overdetermined systems of homogeneous convolution equations on the real line. One of the main topics of the first section is a local version of the Schwartz theorem on spectral analysis. In other sections mean periodic functions with respect to a couple distributions T₁,T₂ are investigated. New effects are discovered: for example, in many cases an important role is played by the rate at which the roots of the Fourier transforms $\widehat{T}_{1}$ and $\widehat{T}_{2}$ “come together” at infinity. Also, connections with division-type formulas for entire functions are discussed. Finally, the deconvolution problem is considered, and some explicit reconstruction formulas are given.