Wavelet frames with irregular matrix dilations and their stability
Stability questions for wavelet frames in \(L^2(\mathbb R)\) have been considered for several years. The present paper concerns similar questions in \(L^2(\mathbb R^n)\), which requires more complicated techniques. A sequence \(\{D_j\}_{j\in \mathbb Z}\) of real nonsingular \(n\times n\) matrices is called an MFS if for any numbers \(a,b\), \(0<a<b<\infty\), there exists a constant \(N\) (depending only on \(b/a\)) such that \(\sharp \{j\in \mathbb Z: a < | D_jw| \leq b\} < N\) for any \(\omega \in \mathbb R^n\setminus \{0\}\). The main result states the following. Assume that \(\{A_j^{-T}:\,j\in \mathbb Z\}\) is an MSF, \(\widehat{\psi}\) is continuous and that: 1) There exists constant \(L>0\) such that \[ L \leq \sum_{j\in \mathbb Z} \left| \widehat{\psi}(A_j^{-T}\omega) \right| ^2, \quad \forall \omega\in \mathbb R^n\setminus\{0\}. \] 2) There exist constants \(C>0, \alpha>0\) and \(\beta>n\) such that \[ | \widehat{\psi}(\omega)| \leq C \min \{ | \omega| ^{\alpha}, | \omega| ^{-\beta}\}, \quad \forall \omega\in \mathbb R^n\setminus \{0\}. \] Then for any \(\gamma\) with \(0<\gamma<\min\{2, \beta\}\) there exist positive constants \(\delta_i \) such that \[ \{| \det S_j| ^{1/2}\psi(S_jx - B\lambda_k): j\in \mathbb Z, k\in \mathbb Z^n\} \] is a frame for \(L^2( \mathbb R^n)\) for any real nonsingular \(n\times n\) matrix sequence \(\{S_j\}_{j\in \mathbb Z}\) satisfying \[ \left\| (S_j^{-T} - A_j^{-T}) \omega \right\| _2 \leq \delta_1 \| A_j^{-T}\omega\| _2, \forall \omega\in \mathbb R^n, \quad j\in \mathbb Z, \] any nonsingular matrix \(B\) with \(\| B\| \leq \delta_2\), and any sequence \(\{\lambda_k\}_{k\in \mathbb Z^d}\subset \mathbb R^n\) with \(\sum_{k\in \mathbb Z^n} \| k -\lambda_k\| _2^{\gamma} < \delta_3/ (| \det B| \cdot \| B\| ^{\gamma})\).
scientific article; zbMATH DE number 2093795