Continuous Partial Gabor Transform for Semi-Direct Product of Locally Compact Groups

Abstract

Let \(H\) be a locally compact group, \(K\) be an LCA group, \(\tau :H\rightarrow Aut(K)\) be a continuous homomorphism and \(G_\tau =H\ltimes _\tau K\) be the semi-direct product of \(H\) and \(K\) with respect to the continuous homomorphism \(\tau \). In this article, we introduce the \(\tau \times \widehat{\tau }\)-time frequency group \(G_{\tau \times \widehat{\tau }}\). We define the \(\tau \times \widehat{\tau }\)-continuous Gabor transform of \(f\in L^2(G_\tau )\) with respect to a window function \(u\in L^2(K)\) as a function defined on \(G_{\tau \times \widehat{\tau }}\). It is also shown that the \(\tau \times \widehat{\tau }\)-continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of \(L^2(G_\tau )\) as a window function. Finally, we indicate some possible applications of these methods in the case of some well-known semi-direct product groups.