Multiresolution Analysis from a Riesz Family of Shifts of a Refinable Function in L²(G)

Abstract

Let G be a second countable locally compact abelian group, L be a uniform lattice in G and $S_L$ be a fundamental domain for L in G. Let $L_{\circ }^p(G)= \{ \varphi : G \longrightarrow {\mathbb {C}};\quad \big \Vert \sum _{k\in L}|\varphi (k^{-1}x)|\big \Vert _{L^p(S_L)} < \infty \}$ $(1\leqslant p\leqslant \infty )$. In this paper we aim among other things, to introduce the Banach space $L_{\circ }^p(G)$ $(1\leqslant p\leqslant \infty )$, with the norm $|\cdot |_p$, and for $p=2$ and a refinable function $\varphi \in L_{\circ }^2(G)$ and the Riesz family generated by the shifts of $\varphi$ by L in G, construct a multiresolution analysis in L²(G). Also some examples are provided to support our construction.