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Brief communications

The unique solvability of one class of multiplicative convolution equations

L. G. Salekhov^a, L. L. Salekhova<sup>b

a</sup> Chair of Differential Equations, Kazan (Volga Region) Federal University, Kazan, Russia
^b Chair of Mathematical Linguistics anf Informational Systems in Philology, Kazan (Volga Region) Federal University, Kazan, Russia

Abstract: In the space of tempered distributions we consider a certain equations on the real axis with operations of convolution and multiplication. It contains convolution equations, in particular, ordinary differential equations with constant coefficients, finite-difference equations, functional differential equations with constant coefficients and shifts, as well as pair differential equations. Owing to the possibility of the analytical representation of distributions (the Cauchy or Hilberts transform), the considered class of equations is equivalent to a certain class of boundary value problems of the Riemann type, where equations play the role of boundary conditions in the sense of tempered distributions. As the research technique we use the Fourier transform, the generalized Fourier transform (the Carleman–Fourier transform), as well as the theory of convolution equations in the space of distributions.

Keywords: convolution equation, convolution algebra and moduli, Fourier transform, Carleman–Fourier transform, analytical representation of distributions, tempered distributions, space of convolutors for the space of tempered distributions.

UDC: 517.982

Presented by the member of Editorial Board: S. G. Samko
Received: 17.05.2011

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2012, 56:5, 57–60

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