Sampling set conditions in weighted multiply generated shift-invariant spaces and their applications

Abstract

In this paper, we investigate the non-uniform weighted average sampling for the weighted multiply generated shift-invariant space. The conditions for sampling point set to be a set of sampling for the weighted multiply generated shift-invariant space are obtained. Furthermore, the explicit bound expressions of a set of sampling inequalities for this shift-invariant spaces also are obtained.

Weighted sampling and reconstruction in weighted reproducing kernel spaces

“…We describe the sampling process (2.21) as weighted sampling, as in the paper the sampling functional ψ γ is supported in a small neighborhood of the location γ for each γ ∈ Γ . The reader may refer to [2,3,9,14,18,19,22,23] for weighted sampling of a band-limited signal, a signal in a shift-invariant space, or a signal with finite rate of innovation.…”

“…where ψ γ to be known as the weighted sampling functional [2,3,9,14,18,19,22,23]. The conclusions are only established in [12] for the ideal sampling and trivial weight case.…”

An analysis of wavelet frame based scattered data reconstruction

“…Besides its structural simplicity shift invariant spaces have the beneficial property that for special choices of φ they provide good approximation orders to sufficiently smooth functions, see [25,26,29]. The compact support of φ also results in sparse system matrices which is of computational interest [29,40]. Such scaling functions φ are for instance B-splines, which in turn give rise to associated wavelet frame systems, as we discuss below.…”

A New Reconstruction Algorithm in Spline Signal Spaces

“…However, this problem is ill-posed, and becomes meaningful only when the function f is assumed to be bandlimited, or to belong to a shift-invariant space [1,2,3,4,8,11,12]. For a bandlimited signal of finite energy, it is completely characterized by its samples, and described by the famous classical Shannon sampling theorem.…”

“…For example, the sampling steps need to be fluctuated according to the signals so as to reduce the number of samples and the computational complexity. If a weighted sampling is considered, the system will be made to be more efficient [1,2,3,4,5,11,12]. It is well known that spline subspaces yield many advantages in their generation and numerical treatment so that there are many practical applications for signal or image processing.…”