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June, 2021 Frame-based Average Sampling in Multiply Generated Shift-invariant Subspaces of Mixed Lebesgue Spaces
Yingchun Jiang, Jiao Li
Taiwanese J. Math. 25(3): 535-552 (June, 2021). DOI: 10.11650/tjm/201002

## Abstract

In this paper, we mainly discuss the nonuniform average sampling and reconstruction in multiply generated shift-invariant subspaces $V_{p,q}(\Phi_r) = \bigg\{ \sum_{k_{1} \in \mathbf{Z}} \sum_{k_{2} \in \mathbf{Z}^{d}} c^T(k_{1},k_{2}) \Phi_r(\,\cdot-k_{1},\,\cdot-k_{2}): (c(k_{1},k_{2}))_{(k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}} \in \big( \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d) \big)^r \bigg\}$ of mixed Lebesgue spaces $L^{p,q}(\mathbf{R} \times \mathbf{R}^{d})$, $1 \leq p,q \leq \infty$, where $\Phi_r = (\varphi_1, \varphi_2, \ldots, \varphi_r)^T$ with $\varphi_i \in L^{p,q}(\mathbf{R} \times \mathbf{R}^d)$ and $c = (c_1,c_2,\ldots,c_r)^T$ with $c_i \in \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d)$, $i = 1,2,\ldots,r$, under the assumption that the family $\{ \varphi_{i}(x-k_{1},y-k_{2}): (k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}, 1 \leq i \leq r \}$ constitutes a $(p,q)$-frame of $V_{p,q}(\Phi_r)$. First, iterative approximation projection algorithms for two kinds of average sampling functionals are established. Then, we estimate the convergence rates of the corresponding algorithms.

## Funding Statement

The project is partially supported by the National Natural Science Foundation of China (No. 11661024) and the Guangxi Natural Science Foundation (Nos. 2020GXNSFAA159076, 2019GXNSFFA245012, 2017GXNSFAA198194), Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS201925), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.

## Citation

Yingchun Jiang. Jiao Li. "Frame-based Average Sampling in Multiply Generated Shift-invariant Subspaces of Mixed Lebesgue Spaces." Taiwanese J. Math. 25 (3) 535 - 552, June, 2021. https://doi.org/10.11650/tjm/201002

## Information

Received: 16 March 2020; Revised: 18 September 2020; Accepted: 14 October 2020; Published: June, 2021
First available in Project Euclid: 20 October 2020

Digital Object Identifier: 10.11650/tjm/201002

Subjects:
Primary: ‎42C40, 94A20

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 3 • June, 2021