On solvability of convolution equations in spaces of generalized distributions with restricted growth
In mathematics, analysis is endowed with an immensely wide variety of texts containing articles on solvability in spaces of generalized distributions for convolution equations. For the restricted growth, the author has textured the text in a much lucid and impressive form. The present article examines, in a systematic manner, some of the recent works and has employed them in proving the necessary and sufficient conditions for the solvability of convolution equations in spaces of generalized distributions with restricted growth. The main result of this paper is the theorem, which under certain suitable prescribed conditions, states that, if \(u\in{\mathcal O}_c'\) (\(K_{M,w}'\); \(K_{M,w}'\)) and \(s*u\in K_{M,w}\), then \(u\in K_{M,w}\) and other conditions are equivalent for a generalized distribution \(S\). Using the Paley-Wiener theorem, applied to the Fourier transform and to the element of \(K_{M,w}\), the said theorem is established. The importance of this paper lies in the fact, that, it involves the topology of the space of convolution operators \(K_{M,w}'\) with restricted exponential growth.
solvability in spaces of generalized distributions
scientific article; zbMATH DE number 1289963