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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 $$1 < p < \infty$$.

##### MSC:
 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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