Convolution random sampling in multiply generated shift-invariant spaces of $L^p(\mathbb {R}^{d})$

Abstract

We mainly consider the stability and reconstruction of convolution random sampling in multiply generated shift-invariant subspaces

$$\begin{aligned} V^{p}(\varPhi )=\left\{ \sum \limits _{k\in \mathbb {Z}^{d}}c(k)^{T}\varPhi (\cdot -k):(c(k))_{k\in \mathbb {Z}^{d}}\in (\ell ^{p}(\mathbb {Z}^{d}))^r \right\} \end{aligned}$$

of $L^p(\mathbb {R}^{d})$, $1<p<\infty$, where $\varPhi =(\phi _{1},\phi _{2},\ldots ,\phi _{r})^{T}$ with $\phi _{i}\in L^{p}(\mathbb {R}^{d})$ and $c=(c_{1},c_{2},\ldots , c_{r})^{T}$ with $c_{i}\in \ell ^{p}(\mathbb {Z}^{d})$, $i=1,2,\ldots , r$. The sampling set $\{x_j\}_{j\in \mathbb {N}}$ is randomly chosen with a general probability distribution over a bounded cube $C_{K}$ and the samples are the form of convolution $\{f*\psi (x_j)\}_{j\in \mathbb {N}}$ of the signal f. Under some proper conditions for the generator $\varPhi$, convolution function $\psi$ and probability density function $\rho$, we first approximate $V^{p}(\varPhi )$ by a finite dimensional subspace

$$\begin{aligned} V^{p}_{N}(\varPhi )=\left\{ \sum \limits _{i=1}^{r}\sum \limits _{|k|\le N}c_{i}(k)\phi _{i}(\cdot -k): c_{i}\in \ell ^{p}([-N,N]^{d})\right\} . \end{aligned}$$

Then we show that the sampling stability holds with high probability for all functions in certain compact subsets

$$\begin{aligned} V^{p}_{K}(\varPhi )=\left\{ f\in V^{p}(\varPhi ):\int _{C_{K}}|f*\psi (x)|^{p}dx\ge (1-\delta )\int _{\mathbb {R}^{d}}|f*\psi (x)|^{p}dx\right\} \end{aligned}$$

of $V^{p}(\varPhi )$ when the sampling size is large enough. Finally, we prove that the stability is related to the properties of the random matrix generated by $\{\phi _i*\psi \}_{1\le i\le r}$ and give a reconstruction algorithm for the convolution random sampling of functions in $V^{p}_N(\varPhi )$.