Frames in Hermite-Bergman and special Hermite-Bergman spaces

Abstract

It is well known that the image of \(L^2(\mathbb {R}^n)\) under the Hermite semigroup \(e^{-tH}\) is a Bergman space \(\mathcal {H}_t(\mathbb {C}^n)\) with the reproducing kernel \(L_t\), for \(t>0\). In this paper, it is shown that the discrete system \(\{e_{ap+ibq, t}:p,q\in \mathbb {Z}^n\}\), for \(a,b>0\), arising out of \(L_t\) turns out to be a frame for \(\mathcal {H}_t(\mathbb {C}^n)\) iff \(ab<\pi \sinh 4t\). A similar type of problem is also discussed for Bessel sequences and frames in holomorphic Sobolev spaces associated with the Hermite semigroup. An attempt is also made to study a similar type of problem for the Bergman space \(\mathcal {B}_t^*(\mathbb {C}^{2n})\), which is the image of \(L^2(\mathbb {C}^n)\) under the special Hermite semigroup.