Hilbert space valued Gabor frames in weighted amalgam spaces

TL;DR

This paper extends the theory of Gabor frames to Hilbert space-valued functions in weighted amalgam spaces, establishing representations and invertibility of the frame operator, including for superframes and multi-window frames.

Contribution

It generalizes Walnut's and Janssen's representations for $ ext{H}$-valued Gabor frames in weighted amalgam spaces and proves their invertibility under certain conditions.

Findings

Established Walnut and Janssen representations for $ ext{H}$-valued Gabor frame operators.

Proved invertibility of the frame operator on weighted amalgam spaces under specific window conditions.

Extended results to Gabor superframes and multi-window Gabor frames.

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 $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty},L^1_w)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty,$ as a special case by choosing the appropriate Hilbert space $\mathbb{H}$.