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

On s-Elementary Super Frame Wavelets and Their Path-Connectedness A super wavelet of length n is an n-tuple (ψ 1,ψ 2,…,ψ 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 (η 1,η 2,…,η 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 τ-disjoint if the E j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E 1,E 2,…,E m ) of frame sets (i.e., η j defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ is a frame wavelet for L 2(ℝ) for each j) lead to a super frame wavelet (η 1,η 2,…,η 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 (η 1,η 2,…,η 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(ℝ) with frame bound k 0 if and only if each η j is a tight frame wavelet for L 2(ℝ) with frame bound k 0 and that (E 1,E 2,…,E m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏1≤j≤m L 2(ℝ) 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(ℝ) 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(ℝ) norm, for any given positive integers m and k 0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

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

Acta Applicandae Mathematicae , Volume 116 (2) – Jul 26, 2011

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References (24)

Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Science+Business Media B.V.
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Computer Science, general; Mechanics; Mathematics, general; Statistical Physics, Dynamical Systems and Complexity
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-011-9635-5
Publisher site
See Article on Publisher Site

Abstract

A super wavelet of length n is an n-tuple (ψ 1,ψ 2,…,ψ 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 (η 1,η 2,…,η 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 τ-disjoint if the E j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E 1,E 2,…,E m ) of frame sets (i.e., η j defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ is a frame wavelet for L 2(ℝ) for each j) lead to a super frame wavelet (η 1,η 2,…,η 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 (η 1,η 2,…,η 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(ℝ) with frame bound k 0 if and only if each η j is a tight frame wavelet for L 2(ℝ) with frame bound k 0 and that (E 1,E 2,…,E m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏1≤j≤m L 2(ℝ) 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(ℝ) 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(ℝ) norm, for any given positive integers m and k 0.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Jul 26, 2011

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