Random sampling in shift invariant spaces

Abstract

a b s t r a c tThe set of sampling in a shift invariant space plays an important role in signal processing and has many applications. This paper addresses the problem when some randomly chosen samples X = {x j : j ∈ J} form a set of sampling in a shift invariant space.holds uniformly for all functions f in a shift invariant space, where c p and C p are positive constants (1 ≤ p ≤ ∞). We prove that with overwhelming probability, the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large.

Random average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue spaces

“…They obtained the probabilistic sampling inequality for band-limited functions on R in [3] and the same for band-limited functions on R d in [4]. Random sampling in shift-invariant spaces was studied in [26,24,9]. Yang and Tao in [25] studied random sampling and gave an approximation model for signals having bounded derivatives.…”

Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $L^p({\mathbb R}^n)$

“…Bass and Gröchenig studied random sampling for multivariate trigonometric polynomial [2]; Candés, Romberg, and Tao reconstructed sparse trigonometric polynomial from a random sample set [6]. In the last decades, random sampling studied for Paley-Wiener space [3,4]; shift-invariant space [15,33,35]; continuous function space with bounded derivative [34]; function space with finite rate of innovation [24]; reproducing kernel subspace of L p (R n ) which is an image of an idempotent integral operator [23,27].…”

Relevant sampling in finitely generated shift-invariant spaces

“…There exist several well-understood ways of formulating these restrictions, eg. by assuming the analog signal f to be bandlimited, or more generally, that it belongs to a shift-invariant space [1,2,3,5,6,10,11,15,16,18,19,20]. Bandlimited signals of finite energy are completely characterized by their regular samples if they are taken at a sufficiently high rate (Nyquist criterion), as described by the famous classical Shannon sampling theorem.…”

“…Recently, the random sampling problems were studied by Bass and Gröchenig in the multivariate trigonometric polynomials spaces [4] and bandlimited functions spaces [5,6]. Yang and Wei discussed the problem when some randomly chosen samples X = {x j : j ∈ J} forms a set of sampling in the shift-invariant space [20].…”