Frames in Hermite-Bergman and special Hermite-Bergman spaces
Let \(\mathcal H_t(\mathbb C^n)\), \(t>0\), denote the space of those entire functions \(F\) on \(\mathbb C^n\) for which \[ \| F\|^2_{\mathcal H_t}=\int_{\mathbb R^{2n}}| F(x+ iy)|^2 U_t (x+ iy)\, dx dy<\infty, \] where \(U_t(x+ iy) = 2^n (\sinh 4t)^{-\frac{n}{2}} \exp(x^2 \tanh 2t - y^2 \coth 2t)\). The Hermite semigroup \(e^{-tH}: L^2(\mathbb R^n) \to \mathcal H_t (\mathbb C^n)\) is known to be an isometric isomorphism. Let \(L_t (z,w)\) denote the reproducing kernel of \(\mathcal H_t(\mathbb C^n)\). Put \[ e_{w,t}(z) = \frac{L_t (z,w)}{\| L_t (\cdot,w)\|_{\mathcal H_t}}, \quad z,w\in\mathbb C^n,\, t>0. \] The authors prove that the discrete system \(\{e_{ap+ibq, t}: p,q\in \mathbb Z^n\}\) with \(a\), \(b>0\) is a frame for \(\mathcal H_t(\mathbb C^n)\) if and only if \(ab<\pi\sinh 4t\). Also, they consider a similar problem for Bessel sequences and frames in holomorphic Sobolev spaces associated with the Hermite semigroup.
scientific article; zbMATH DE number 6818254
scientific article