Hilbert space valued Gabor frames in weighted amalgam spaces

Abstract

Let $\mathbb{H}$ be a separable Hilbert space. In this paper we establish a generalization of Walnut's representation and Janssen's representation of the $\mathbb{H}-$valued Gabor frame operator on $\mathbb{H}-$valued weighted amalgam spaces $W_{\mathbb{H}}(L^p,L_v^q)$, $1\le p,q\le \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L_v^q)$, $1\le p,q\le \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty },L_w^1)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on $W_{\mathbb{H}}(L^p,L_v^q)$, $1\le p,q\le \infty ,$ as a special case by choosing the appropriate Hilbert space $\mathbb{H}$.

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