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

In this article, we consider the random sampling in the image space V$$ V $$ of an idempotent integral operator on mixed Lebesgue space Lp,qℝn+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. [ABSTRACT FROM AUTHOR]