On the inclusion relations between Gelfand–Shilov spaces

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 in one another in terms of growth relations for the defining weight sequence and weight 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–Björck spaces $\mathcal {S}^{[\omega ]}_{[\eta ]}$ (defined via weight functions $\omega $ and $\eta $).