On Gelfand--Shilov spaces of type $S$

I. Kh. Musin

Vladikavkazskij matematičeskij žurnal, Tome 27 (2025) no. 1, pp. 87-100

Résumé

In the theory of generalized functions and the theory of differential equations spaces of rapidly decreasing infinitely differentiable functions are of considerable interest. This is due to the fact that when solving various problems of analysis in such spaces one can use the rich possibilities provided by the Fourier transform or the Laplace transform. One of such spaces is the Gelfand–Shilov spaces of type $S$. They arose in the mid-1950s in the works of I. M. Gelfand and G. E. Shilov during the study of the problem of uniqueness of the solution of Cauchy problems for partial differential equations. In the famous series of books by I. M. Gelfand and G. E. Shilov on generalized functions of the late 1950s – early 1960s the properties of the functions of these spaces are described in detail and a thorough Fourier analysis is carried out in them. By now, spaces of type $S$ have found numerous applications also in the theory of pseudodifferential operators, time-frequency analysis. In the present paper, using two countable families ${\varphi}$ and $\psi$ of separately radial weight functions in ${\mathbb R}^n$, we introduce a space ${\mathcal S}_{\varphi}^{\psi}$ of functions of type $S$ that is more general than the Gelfand–Shilov space $S_{\alpha}^{\beta}$. We obtain a description of the space ${\mathcal S}_{\varphi}^{\psi}$ in terms of the Fourier transform of functions and consider the question of its non-triviality. The study of the periodization operator on one of the spaces of type $S$ under consideration turned out to be related to the problem of describing the functions of the space of periodic ultradifferentiable functions of Roumieu type in terms of the decrease of their Fourier coefficients.