Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in

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@Article{ATA-31-221,
author = {Z. Li and X. Xu},
title = {Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in

*L*}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {3}, pages = {221--235}, abstract = {The single 2 dilation orthogonal wavelet multipliers in onedimensional case and single^{2}(R^{2})*A*-dilation (where*A*is any expansivematrix with integer entries and*|detA|=2*) wavelet multipliers inhigh dimensional case were completely characterized by the WutamConsortium (1998) and Z. Y. Li, et al. (2010). But there exist no moreresults on orthogonal multivariate wavelet matrix multiplierscorresponding integer expansive dilation matrix with the absolutevalue of determinant not 2 in*L*. In this paper, wechoose $$2I_2=\left(\begin{array}{cc}2 & 0\\0 & 2\end{array}\right)$$ as the dilation matrix and consider the^{2}(R^{2})*2I*-dilation orthogonal multivariate wavelet$\Psi=\{\psi_1,\psi_2,\psi_3\}$, (which is called a dyadic bivariatewavelet) multipliers. We call the $3\times 3$ matrix-valued function$A(s)=[f_{i,j}(s)]_{3\times 3}$, where $f_{i,j}$ are measurablefunctions, a dyadic bivariate matrix Fourier wavelet multiplier ifthe inverse Fourier transform of$A(s)(\widehat{\psi_{1}}(s),\widehat{\psi_{2}}(s),\widehat{\psi_{3}}(s))^{\top}=(\widehat{g_1}(s),\widehat{g_2}(s),\widehat{g_3}(s))^{\top}$is a dyadic bivariate wavelet whenever$(\psi_{1},\psi_{2},\psi_{3})$ is any dyadic bivariate wavelet. Wegive some conditions for dyadic matrix bivariate waveletmultipliers. The results extended that of Z. Y. Li and X. L. Shi (2011). As an application, we construct some useful dyadicbivariate wavelets by using dyadic Fourier matrix waveletmultipliers and use them to image denoising.}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n3.1}, url = {http://global-sci.org/intro/article_detail/ata/4635.html} }_{2}
TY - JOUR
T1 - Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in

*L*AU - Z. Li & X. Xu JO - Analysis in Theory and Applications VL - 3 SP - 221 EP - 235 PY - 2017 DA - 2017/07 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n3.1 UR - https://global-sci.org/intro/article_detail/ata/4635.html KW - Multi-wavelets KW - Fourier multipliers KW - image denoising AB - The single 2 dilation orthogonal wavelet multipliers in onedimensional case and single^{2}(R^{2})*A*-dilation (where*A*is any expansivematrix with integer entries and*|detA|=2*) wavelet multipliers inhigh dimensional case were completely characterized by the WutamConsortium (1998) and Z. Y. Li, et al. (2010). But there exist no moreresults on orthogonal multivariate wavelet matrix multiplierscorresponding integer expansive dilation matrix with the absolutevalue of determinant not 2 in*L*. In this paper, wechoose $$2I_2=\left(\begin{array}{cc}2 & 0\\0 & 2\end{array}\right)$$ as the dilation matrix and consider the^{2}(R^{2})*2I*-dilation orthogonal multivariate wavelet$\Psi=\{\psi_1,\psi_2,\psi_3\}$, (which is called a dyadic bivariatewavelet) multipliers. We call the $3\times 3$ matrix-valued function$A(s)=[f_{i,j}(s)]_{3\times 3}$, where $f_{i,j}$ are measurablefunctions, a dyadic bivariate matrix Fourier wavelet multiplier ifthe inverse Fourier transform of$A(s)(\widehat{\psi_{1}}(s),\widehat{\psi_{2}}(s),\widehat{\psi_{3}}(s))^{\top}=(\widehat{g_1}(s),\widehat{g_2}(s),\widehat{g_3}(s))^{\top}$is a dyadic bivariate wavelet whenever$(\psi_{1},\psi_{2},\psi_{3})$ is any dyadic bivariate wavelet. Wegive some conditions for dyadic matrix bivariate waveletmultipliers. The results extended that of Z. Y. Li and X. L. Shi (2011). As an application, we construct some useful dyadicbivariate wavelets by using dyadic Fourier matrix waveletmultipliers and use them to image denoising._{2}
Z. Li & X. Xu. (1970). Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in

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*L*.^{2}(R^{2})*Analysis in Theory and Applications*.*31*(3). 221-235. doi:10.4208/ata.2015.v31.n3.1