Construction of periodic wavelet frames with dilation matrix (Q2258917)

For a normalized tight wavelet frame \(\{\varphi,\psi_1,\dots,\psi_l\}\) for \(L^2(\mathbb{R}^d)\) derived from a refinable function and a filter bank \(\{H_0,H_1,\dots, H_l\}\), under the condition that \(\{\varphi,\psi_1,\dots,\psi_l\}\subset L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)\) and assuming that all these generators have a common radial decreasing \(L^1\)-majorant (see Definition 2.1), the authors establish in Theorem 3.1 that the wavelet system in (3.1) after applying the periodization technique is a normalized periodic tight wavelet frame for \(L^2([0,1]^d)\). Similar results have also been obtained in Theorem 4.1 for constructing periodic dual wavelet frames for \(L_2([0,1]^d)\) derived from a pair of dual frames for \(L^2(\mathbb{R}^d)\) through multiresolution analysis, under the assumption that all the generators \(\varphi,\tilde{\varphi},\psi_1,\dots,\psi_l,\tilde{\psi}_1,\dots,\tilde{\psi}_l\) are compactly supported. The results in this paper not only extend several results in the literature to the general dilation matrix case but also show that the periodization technique for building periodic tight or dual wavelet frames works under a weaker condition than previously known from similar results.