Illinois Journal of Mathematics

The path-connectivity of MRA wavelets in $L^2(\mathbb {R}^{d})$

Zhongyan Li, Xingde Dai, Yuanan Diao, and Wei Huang

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Abstract

We show that for any $d\times d$ expansive matrix $A$ with integer entries and $|\det(A)|=2$, the set of all $A$-dilation MRA wavelets is path-connected under the $L^2(\mathbb{R}^d)$ norm topology. We do this through the application of $A$-dilation wavelet multipliers, namely measurable functions $f$ with the property that the inverse Fourier transform of $(f\widehat{\psi})$ is an $A$-dilation wavelet for any $A$-dilation wavelet $\psi$ (where $\widehat{\psi}$ is the Fourier transform of $\psi$). In this process, we have completely characterized all $A$-dilation wavelet multipliers for any integral expansive matrix $A$ with $|\det(A)|=2$.

Article information

Source
Illinois J. Math., Volume 54, Number 2 (2010), 601-620.

Dates
First available in Project Euclid: 14 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1318598674

Digital Object Identifier
doi:10.1215/ijm/1318598674

Mathematical Reviews number (MathSciNet)
MR2846475

Zentralblatt MATH identifier
1238.42019

Subjects
Primary: 42-XX: HARMONIC ANALYSIS ON EUCLIDEAN SPACES 46-XX: FUNCTIONAL ANALYSIS {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx}

Citation

Li, Zhongyan; Dai, Xingde; Diao, Yuanan; Huang, Wei. The path-connectivity of MRA wavelets in $L^2(\mathbb {R}^{d})$. Illinois J. Math. 54 (2010), no. 2, 601--620. doi:10.1215/ijm/1318598674. https://projecteuclid.org/euclid.ijm/1318598674


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