## 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
Full Text:

### References:

 [1] Bownik, M.: The structure of shift-invariant subspaces of L 2(R n ). J. Funct. Anal. 177(2), 282–309 (2000) · Zbl 0986.46018 [2] Bownik, M.: Inverse volume inequalities for matrix weights. Indiana Univ. Math. J. 50(1), 383–410 (2001) · Zbl 0992.42006 [3] Chang, S.-Y.A., Fefferman, R.: Some recent developments in Fourier analysis and H p -theory on product domains. Bull. Am. Math. Soc. (N.S.) 12(1), 1–43 (1985) · Zbl 0557.42007 [4] Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992) · Zbl 0776.42018 [5] de Boor, C., Ron, A.: Fourier analysis of the approximation power of principal shift-invariant spaces. Constr. Approx. 8(4), 427–462 (1992) · Zbl 0801.41027 [6] de Boor, C., DeVore, R.A., Ron, A.: On the construction of multivariate (pre)wavelets. Constr. Approx. 9(2–3), 123–166 (1993) · Zbl 0773.41013 [7] de Boor, C., DeVore, R.A., Ron, A.: Approximation from shift-invariant subspaces of $$L_2(\mathbf R^ d)$$ . Trans. Am. Math. Soc. 341(2), 787–806 (1994) · Zbl 0790.41012 [8] de Boor, C., DeVore, R.A., Ron, A.: The structure of finitely generated shift-invariant spaces in L 2(R d ). J. Funct. Anal. 119(1), 37–78 (1994) · Zbl 0806.46030 [9] Dyn, N., Ron, A.: Radial basis function approximation: from gridded centres to scattered centres. Proc. Lond. Math. Soc. (3) 71(1), 76–108 (1995) · Zbl 0827.41007 [10] Frazier, M., Roudenko, S.: Matrix-weighted Besov spaces and conditions of A p type for 0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.