Random sampling and reconstruction in reproducing kernel subspace of mixed Lebesgue spaces. (1st November 2022) - Goyal, Prashant; Patel, Dhiraj; Sivananthan, S.
Abstract : In this article, we consider the random sampling in the image space V $$ V $$ of an idempotent integral operator on mixed Lebesgue space L p, q ℝ n + 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 C R, S = − R 2, R 2 n × − S 2, S 2 $$ {C}_{R, S}={\left[-\frac{R}{2}, \frac{R}{2}\right]}^n\times \left[-\frac{S}{2}, \frac{S}{2}\right] $$ . Consequently, we show that the set of bounded functions concentrated on C R, S $$ {C}_{R, S} $$ is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over C R, S $$ {C}_{R, S} $$ is a stable set of sampling for the set of concentrated functions on C R, S $$ {C}_{R, S} $$ . Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.