Abstract
The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L 2(ℝ2). In this paper, we choose \(2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right)\) as the dilation matrix and consider the 2I 2-dilation multivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f 1, f 2, f 3} a dyadic bivariate wavelet multiplier if \(\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\}\) is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}, where \(\hat f\) and F −1 denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.
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Supported by NSFC (Grant Nos. 10671062 and 11071065), Ph. D Programs Foundation of Ministry Education of China (Grant No. 20094306110004); the first author is also partially supported by the Project-sponsored by SRF for ROCS, SEM, the Fundamental Research Funds for the Central Universities, and China Postdoctoral Science Foundation funded project (Grant No. 20100480942)
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Li, Z.Y., Shi, X.L. Dyadic bivariate wavelet multipliers in L 2(ℝ2). Acta. Math. Sin.-English Ser. 27, 1489–1500 (2011). https://doi.org/10.1007/s10114-011-9772-9
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DOI: https://doi.org/10.1007/s10114-011-9772-9
Keywords
- Dyadic bivariate wavelet
- dyadic bivariate wavelet multiplier
- dyadic MRA bivariate wavelet
- dyadic low pass filter
- Haar type dyadic wavelet