Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group pp 575-596 | Cite as

# 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 *T*_{1},*T*_{2} 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.

## Keywords

Entire Function Periodic Function Heisenberg Group Select Versus Convolution Equation## Preview

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