Relevant sampling in a reproducing kernel subspace of Orlicz space

Abstract

In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space [Formula: see text]. We consider the function space [Formula: see text] as the image of an idempotent integral operator on [Formula: see text], where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of [Formula: see text] includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in [Formula: see text] can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in [Formula: see text] concentrated on a cube [Formula: see text], when the samples collected at i.i.d. random points are drawn on [Formula: see text] of order [Formula: see text].