A New Look at the Schwartz Theory

  • Valery V. VolchkovEmail author
  • Vitaly V. Volchkov
Part of the Springer Monographs in Mathematics book series (SMM)


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.


Entire Function Periodic Function Heisenberg Group Select Versus Convolution Equation 
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Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  1. 1.Mathematical DepartmentDonetsk National UniversityDonetskUkraine

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