Characterization of matrix Fourier multipliers for \(A\)-dilation Parseval multi-wavelet frames (Q2788484)

Let \(A\) be an \(n\times n\) expansive integral matrix with \(|\det(A)|=2\). An \(A\)-dilation Parseval frame wavelet is a function \(\varphi \in L^{2}(\mathbb{R}^{n})\), such that the set \(\{|\det(A)|^{\frac{j}{2}}\varphi(A^{j}.-k):j\in \mathbb{Z},k\in\mathbb{Z}^{n}\}\) forms a Parseval frame for \(L^{2}(\mathbb{R}^{n})\).NEWLINENEWLINEFrame wavelet multipliers were introduced by \textit{M. Paluszyński} et al. [J. Geom. Anal. 11, No. 2, 311--342 (2001; Zbl 0985.42020)] in the one-dimensional case and for the high-dimensional case by \textit{D. Bakić} et al. [Appl. Comput. Harmon. Anal. 19, No. 3, 386--431 (2005; Zbl 1090.42020)] and \textit{Z. Li} and \textit{X. Shi} [Chin. Ann. Math., Ser. B 33, No. 6, 949--960 (2012; Zbl 1259.42023)] while Parseval multi-wavelet frame multipliers in the one-dimensional case were introduced by \textit{Z. Li} and \textit{D. Han} [Appl. Comput. Harmon. Anal. 35, No. 3, 407--418 (2013; Zbl 1294.42005)].NEWLINENEWLINEIn this paper, the authors investigate Parseval multi-wavelet frame multipliers in the high-dimensional case for the matrix \(A\) by using the above results. Also, necessary and sufficient conditions of multi-wavelet frames have been established with several numerical examples. The results are very interesting and useful for researchers working in wavelet analysis.