The unique solvability of one class of multiplicative convolution equations

L. G. Salekhov ; L. L. Salekhova

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2012), pp. 67-71

Résumé

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.

Détail

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