Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces

Abstract

We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, $\mathcal{H}$, and its dual, $\mathcal{H}^\prime$ . We prove that for such quasi-Banach space $\mathcal{B}$, there are convenient Hilbert spaces, $\mathcal{H}_{k}, k=1,2$, with normalized Hermite functions as orthonormal bases and such that $\mathcal{B}$ lies between $\mathcal{H}_1\; \mathrm{and}\;\mathcal{H}_2$, and the latter spaces lie between $\mathcal{H}\; \mathrm{and}\;\mathcal{H}^\prime$.