On s-Elementary Super Frame Wavelets and Their Path-Connectedness

Abstract

A super wavelet of length n is an n-tuple (? 1,? 2,?,? n ) in the product space $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ , such that the coordinated dilates of all its coordinated translates form an orthonormal basis for $\prod_{j=1}^{n} L^{2} (\mathbb{R})$ . This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (? 1,? 2,?,? n ) in $\prod_{j=1}^{n}L^{2} (\mathbb{R})$ such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ . In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E 1,E 2,?,E m ) is said to be ?-disjoint if the E j 's are pair-wise disjoint under the 2?-translations. We prove that a ?-disjoint m-tuple (E 1,E 2,?,E m ) of frame sets (i.e., ? j defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ is a frame wavelet for L 2(?) for each j) lead to a super frame wavelet (? 1,? 2,?,? m ) for $\prod_{j=1}^{m} L^{2} (\mathbb{R})$ where $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ . In the case of super tight frame wavelets, we prove that (? 1,? 2,?,? m ), defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ , is a super tight frame wavelet for ?1?j?m L 2(?) with frame bound k 0 if and only if each ? j is a tight frame wavelet for L 2(?) with frame bound k 0 and that (E 1,E 2,?,E m ) is ?-disjoint. Denote the set of all ?-disjoint s-elementary super frame wavelets for ?1?j?m L 2(?) by $\mathfrak{S}(m)$ and the set of all s-elementary super tight frame wavelets (with the same frame bound k 0) for ?1?j?m L 2(?) by $\mathfrak{S}^{k_{0}}(m)$ . We further prove that $\mathfrak{S}(m)$ and $\mathfrak{S}^{k_{0}}(m)$ are both path-connected under the ?1?j?m L 2(?) norm, for any given positive integers m and k 0.