Sequence space representations of Beurling-Björck spaces via Gabor frames and Wilson bases

Abstract

We establish sequence space representations of a broad class of Beurling-Björck spaces ${\mathcal{S}}_{(\eta )}^{(\omega )}$ and ${\mathcal{S}}_{\{\eta \}}^{\{\omega \}}$. We develop two different approaches: a non-constructive one based on Gabor frames and the structure theory of Fréchet 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 ${\mathcal{S}}_{(\eta )}^{(\omega )}$ and ${\mathcal{S}}_{\{\eta \}}^{\{\omega \}}$ in terms of $\omega$ and $\eta$. In particular, our results are applicable to the classical Gelfand-Shilov spaces ${\mathcal{S}}_{\tau }^{\mu }$ for $\mu ,\tau \ge 1/2$ (non-constructive approach) and $\mu ,\tau \ge 1$ (constructive approach).

17 pages