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
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