Sequence space representations of Beurling-Bj\"orck spaces via Gabor frames and Wilson bases

Abstract

We establish sequence space representations of a broad class of Beurling-Bj\"orck spaces $\mathcal{S}^{(\omega)}_{(\eta)}$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$. We develop two different approaches: a non-constructive one based on Gabor frames and the structure theory of Fr\'echet spaces, and a constructive one using Wilson bases, under stronger assumptions on the defining weight functions $\omega$ and $\eta$. As an application, we provide an isomorphic classification of the spaces $\mathcal{S}^{(\omega)}_{(\eta)}$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$ in terms of $\omega$ and $\eta$. In particular, our results are applicable to the classical Gelfand-Shilov spaces $\mathcal{S}^\mu_\tau$ for $\mu, \tau \geq 1/2$ (non-constructive approach) and $\mu, \tau \geq 1$ (constructive approach).