On stability of finitely generated shift-invariant systems
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 \(\;f \in S({\Psi})\).
vector Hunt--Muckenhoupt-Wheeden theorem
scientific article; zbMATH DE number 5838815
scientific article