Random sampling and reconstruction in reproducing kernel subspace of mixed Lebesgue spaces

Abstract

In this article, we consider the random sampling in the image space V$$ V $$ of an idempotent integral operator on mixed Lebesgue space Lp,q()ℝn+1$$ {L}^{p,q}\left({\mathbb{R}}^{n+1}\right) $$. We assume some decay and regularity conditions on the integral kernel and show that the bounded functions in V$$ V $$ can be approximated by an element in a finite‐dimensional subspace of V$$ V $$ on CR,S=[]−R2,R2n×[]−S2,S2$$ {C}_{R,S}={\left[-\frac{R}{2},\frac{R}{2}\right]}^n\times \left[-\frac{S}{2},\frac{S}{2}\right] $$. Consequently, we show that the set of bounded functions concentrated on CR,S$$ {C}_{R,S} $$ is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over CR,S$$ {C}_{R,S} $$ is a stable set of sampling for the set of concentrated functions on CR,S$$ {C}_{R,S} $$. Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.

Random average sampling and reconstruction in a reproducing kernel space

“…In the aforementioned papers, the construction of sampling stability strongly depends on two main tools, namely, covering number of signal space and large deviation inequality for a sequence of random variables. In fact, this is the case in most of the references on random sampling, such as previous research [10,11,18,19,[21][22][23][24][25][26]. The sampling stability can also be established based on the analysis of some localization operator and matrix Bernstein inequality; see previous studies [7,9,20] where the number of random samples to ensure sampling inequalities hold with high probability is at least of the order D ln D with D being the size of sampling domain.…”

“…random positions. In the last decade, random sampling has been investigated for bandlimited signals [7], signals in (quasi) shift-invariant spaces [9][10][11][18][19][20][21], signals with finite rate of innovation [22,23], and signals in reproducing kernel spaces [8,[24][25][26]. These papers establish sampling inequalities that hold with high probability when the number of sampling data is large enough.…”

“…These papers establish sampling inequalities that hold with high probability when the number of sampling data is large enough. Some of them also study the reconstruction from the exact sampling data by using sampling expansions or iterative algorithms [8, 23, 24] and from the convolution random samples by using sampling expansions [18, 19, 21]. In Section 4, we study the case that the sampling data are of the form (), and the sampling set $$$ {\Gamma}_{\Omega} $$$ consists of i.i.d.…”

Grid-Based Decimation for Wavelet Transforms with Stably Invertible Implementation

“…The invertibility or stability of the sampled representation is not investigated, however. In the context of random sampling, these properties are considered in the literature on relevant sampling, introduced by Bass and Gröchenig for bandlimited functions [28] and later generalized to various settings [29]- [31], including time-frequency representations [32]. Relevant sampling provides a probabilistic framework for stable sampling of functions that are localized in a domain of finite volume, e.g., bandlimited signals that have only negligible energy outside a finite interval.…”