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On stability of finitely generated shift-invariant systems. (English) Zbl 1209.41006

Let \({\Psi}=\{\psi_1, \dots, \psi_N\} \subset L^2(\mathbb R^d)\), \(T({\Psi}) := \{ \psi(\cdot - k): \psi \in {\Psi}, k \in\mathbb Z^d\}\), and let \(S({\Psi})\) denote the closure of the linear span of \(T({\Psi}\)).
The author focuses on the case where \(T({\Psi})\) has a unique biorthogonal system \(\{g^{\psi}\}\) in \(S({\Psi})\), defines the “rectangular” partial sum operators by \[ T_{\mathbf N}f:= \sum_{\psi \in {\Psi}} \sum_{k\in\mathbb Z^d: |k_i|\leq N_i} \langle f, g_k^{\psi}\rangle \psi(\cdot - k), \] for \(f \in S({\Psi})\) and \({\mathbf N}=(N_1, N_2, \dots, N_d) \in \mathbb{N}_0^d\), where \(\mathbb{N}_0\) denotes the set of nonnegative integers, and finds necessary and sufficient conditions for \(T_{\mathbf N}f\to f\), as \(\min_i N_i \to +\infty\), for all \(\;f \in S({\Psi})\).

MSC:

41A45 Approximation by arbitrary linear expressions
42C15 General harmonic expansions, frames
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References:

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