Frame Multiresolution Analysis of Band-limited Functions over Locally Compact Abelian Groups

Abstract

In this article, we introduce the notion of frame multiresolution analysis (FMRA) of band-limited functions on locally compact Abelian (LCA) groups and derive certain conditions under which the subspaces $V_j=\overline{\text{span}}\big\{D^jT_\lambda\phi:\lambda\in\Lambda\big\},\,j\in\mathbb{Z}$, where $\hat\phi(\omega)=\chi_K(\omega), \omega\in \widehat G, K\subset \widehat G$, constitutes an FMRA for $L^2(G)$. Subsequently, we construct the corresponding wavelet frames of both the dyadic and arbitrary dilations on LCA groups. Nevertheless, all the results are braced with illustrative examples.