On the inclusion relations between Gelfand-Shilov spaces

TL;DR

This paper investigates the inclusion relations among various Gelfand-Shilov spaces, providing a unified framework to characterize when one such space is contained within another based on growth conditions of defining weights.

Contribution

It offers a comprehensive characterization of inclusion relations between Gelfand-Shilov spaces using growth conditions on weight sequences and functions, unifying different space definitions.

Findings

Characterization of inclusion relations via growth conditions.

Unified treatment of Gelfand-Shilov and Beurling-Björck spaces.

General framework applicable to various space definitions.

Abstract

We study inclusion relations between Gelfand-Shilov type spaces defined via a weight (multi-)sequence system, a weight function system, and a translation-invariant Banach function space. We characterize when such spaces are included into one another in terms of growth relations for the defining weight sequence and function systems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{[M]}_{[A]}$ (defined via weight sequences $M$ and $A$) and the Beurling-Bj\"orck spaces $\mathcal{S}^{[\omega]}_{[\eta]}$ (defined via weight functions $\omega$ and $\eta$).