A family of Banach algebras of Gabor atoms

A one-parameter family \(\{\mathcal {E}_{a}: a \in \mathbb {R}^+ \}\) $\{{\mathcal{E}}_a:a\in {\mathbb{R}}^{+}\}$ of function spaces on \(\mathbb {R}\) $\mathbb{R}$ , which are Banach algebras with respect to convolution and pointwise products, is studied. All these spaces are strictly larger than the Wiener space and share many of its nice invariance properties. The spaces \(\mathcal {E}_{a}\) ${\mathcal{E}}_a$ and \(\mathcal {E}_{b}\) ${\mathcal{E}}_b$ coincide if and only if a/b is rational. Many functions in \(\mathcal {E}_{a}\) ${\mathcal{E}}_a$ that are not in the Wiener class but generate Gabor frames in \(L^2(\mathbb {R})\) $L^2(\mathbb{R})$ are constructed.