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Frames associated with expansive matrix dilations. (English) Zbl 1044.42030
Let \(A\) be a real \(n \times n\) matrix which is expansive, i.e., the modulus of each of its eigenvalues is greater than 1. A family \(\{ \phi_\gamma: \gamma \in \Gamma\}\) in \(\dot{\mathbf{F}}_p^{\alpha, q} (\mathbb{R}^n, A, dx) \cap ( \dot{\mathbf{F}}_p^{\alpha,q} (\mathbb{R}^n, A, dx))^*\) is called a frame for the anisotropic Triebel-Lizorkin space \(\dot{\mathbf{F}}_p^{\alpha,q} (\mathbb{R}^n, A, dx)\), if the frame operator \(\mathcal{F}: f \mapsto \sum_{\gamma \in \Gamma} \langle f, \phi_\gamma \rangle \phi_\gamma\) is bounded and invertible on \(\dot{\mathbf{F}}_p^{\alpha, q} (\mathbb{R}^n, A, dx)\). Then if \(B\) is a non-singular \(n\times n\) matrix, it is shown that for certain functions \(\varphi\) satisfying the modified Calderón reproducing formula, there exists \(\eta_0\) such that, for any fixed \(\eta < \eta_0\), the wavelet type system \(\{ \varphi_{j,k}(x) = | \det A| ^{j/2} | \det A| ^\eta \varphi (A^\eta A^j x - A^\eta B k) : j \in \mathbb{Z}, k \in \mathbb{Z}^n \}\) is a frame for \(\dot{\mathbf{F}}_p^{\alpha, q} (\mathbb{R}^n, A, dx)\). The smoothness properties of the associated frame operator are studied and it is shown that the corresponding frame expansion converges almost surely for every \(f \in L^p\), if \(1 < p < \infty\).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
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