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 $\mathcal{S}^{(ω)}_{(η)}$ and $\mathcal{S}^{\{ω\}}_{\{η\}}$. 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 $ω$ and $η$. As an application, we provide an isomorphic classification of the spaces $\mathcal{S}^{(ω)}_{(η)}$ and $\mathcal{S}^{\{ω\}}_{\{η\}}$ in terms of $ω$ and $η$. In particular, our results are applicable to the classical Gelfand-Shilov spaces $\mathcal{S}^μ_τ$ for $μ, τ\geq 1/2$ (non-constructive approach) and $μ, τ\geq 1$ (constructive approach).