Frame-based Average Sampling in Multiply Generated Shift-invariant Subspaces of Mixed Lebesgue Spaces

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