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(R^n+1) . 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=[-R2, R2]^n× [-S2, S2] . 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.