On the structure and interpolation properties of quasi shift-invariant spaces

Abstract

The structure of certain types of quasi shift-invariant spaces, which take the form V (ψ, X ) := span L 2 {ψ(· − x j ) : j ∈ Z} for an infinite discrete set X = (x j ) ⊂ R is investigated. Additionally, the relation is explored between pairs (ψ, X ) and (φ, Y) such that interpolation of functions in V (ψ, X ) via interpolants in V (φ, Y) solely from the samples of the original function is possible and stable. Some conditions are given for which the sampling problem is stable, and for which recovery of functions from their interpolants from a family of spaces V (φα, Y) is possible.

Determination of compactly supported functions in shift-invariant space by single-angle Radon samples

“…In what follows we recall the definition of positive definite functions which have been extensively applied to scattered data interpolation, approximation theory and harmonic analysis (e.g. [12,18,22,38]). We say that a function φ : R d −→ C is positive definite if for all N ∈ N, all sets X = {x 1 , x 2 , .…”

“…If X = Z 2 then V (ϕ, X ) degenerates to a SIS. As implied in [12], the recovery for the functions in V (ϕ, X ) (X = Z 2 ) is much more complicated than that for the SIS. For such a recovery, by [12, section 3.1(A1)] it is required that ϕ is positive definite.…”

Random Average Sampling and Reconstruction in Shift-Invariant Subspaces of Mixed Lebesgue Spaces

“…The theory of mixed Lebesgue spaces has been elaborated by Benedek and Panzone in [7]. The concept of quasi shift-invariance has been considered for analysing sampling and interpolation properties in [3,16,17]. We organize the paper as follows.…”

Regular families of kernels for nonlinear approximation

“…In recent works, these have been called quasi shift-invariant spaces, e.g. [3,15,20,24]. For a more extensive history, Wendland's text on scattered-data interpolation is an excellent reference [45].…”