Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions

Abstract

A Riesz-basis sequence for L 2 [−π, π] is a strictly increasing sequence X := (x j ) j∈Z in R such that the set of functions  e −i x j (·)  j∈Z is a Riesz basis for L 2 [−π, π]. Given such a sequence and a parameter 0 < h ≤ 1, we consider interpolation of functions g ∈ W k 2 (R) at the set (hx j ) j∈Z via translates of the Gaussian kernel. Existence is shown of an interpolant of the formwhich is continuous and square-integrable on R, and satisfies the interpolatory condition I h X (g)(hx j ) = g(hx j ), j ∈ Z. Moreover, the use of the parameter h gives approximation rates of order h k . Namely, there is a constant independent of g such that ∥I h X (g) − g∥ L 2 (R) ≤ Ch k | g | W k 2 (R) . Interpolation using translates of certain functions other than the Gaussian, so-called regular interpolators, is also considered and shown to exhibit the same approximation rates.

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

“…Consequently, Theorem 5 recovers Theorem 1 of [45], which constitutes the special case when ψ is the sinc function, whose Fourier transform is the characteristic function of T. Also regard that the proof follows from Proposition 3(i) in this case given the assumptions on ψ. Examples of regular interpolators may be found in Section 5 of [45] and Section 8 of [28], but here we note that a prominent example is the family of Gaussian generators: φ α (x) = e −|x/α| 2 , α ≥ 1.…”

Nonuniform sampling and recovery of bandlimited functions in higher dimensions

“…Finally, to conclude boundedness, simply notice from the periodization argument above, Lemma 5.1, Plancherel's Identity, and (8), that…”

Stability and Robustness of RBF Interpolation

“…To wit, let f ∈ W k 2 (R) with supp(f ) ⊂ [−1, 1] (this choice of interval is arbitrary for ease of presentation, and can easily be dilated). Then for N ∈ N, the interpolant I N −1 Z f is actually interpolating f at the sequence { j N } N j=−N , and has the following form by combining (5) and (8):…”

“…Specifically, [3,10] contain higher dimensional analogues of Theorem 2.5, while the uniform results in higher dimensions may be found throughout the work of Riemenschneider and Sivakumar. Similarly, extensions to Theorem 4.1 are discussed in [8,22], though these are for specific CIS in higher dimensions which are Cartesian products of univariate ones. Unfortunately, the multivariate case of Theorem 4.6 will likely prove more difficult for the reason that finding CISs in higher dimensions remains elusive even for simple domains.…”

“…Even the first part of the classical sampling question leaves some deep open questions in this area and has seen links with many interesting realms of mathematics including space-tiling, convex geometry, basis theory, and abstract harmonic analysis. Of interest to this work are those nonuniform sampling methods which use RBFs [3,8,10,15,22]. For a survey of some of these themes using multiquadrics consult [9], of which this article is a continuation.…”