The metaplectic action on modulation spaces

Abstract

We study the mapping properties of metaplectic operators $\hat{S}\in \mathrm{Mp}(2d,\mathbb{R})$ on modulation spaces of the type ${\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)$. Our main result is a full characterisation of the pairs $(\hat{S},{\mathrm{M}}^{p,q}({\mathbb{R}}^d))$ for which the operator $\hat{S}:{\mathrm{M}}^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}^{p,q}({\mathbb{R}}^d)$ is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that $\hat{S}$ is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of $\hat{S}:{\mathrm{M}}^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}^{p,q}({\mathbb{R}}^d)$ transfers to $\hat{S}:{\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)$.

24 pages, 2 figures. To appear in Appl. Comput. Harmon. Anal

References in corpus (1)

• On foundational discretization barriers in STFT phase retrieval

Cited by in corpus (1)

• Boundedness of metaplectic operators within $L^p$ spaces, applications to pseudodifferential calculus, and time-frequency representations