Random sampling and reconstruction in reproducing kernel subspace of mixed Lebesgue spaces. (1st November 2022)

• Record Type: Journal Article
• Authors: Goyal, Prashant; Patel, Dhiraj; Sivananthan, S.
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 5(2023)
• Page Start: 5119
• Page End: 5138
• Publication Date: 2022-11-01
• Subjects: mixed Lebesgue space -- random sampling -- reconstruction algorithm -- reproducing kernel space; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8821
• Languages: English
• ISSNs: 0170-4214

Abstract

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.