Nonuniform sampling and recovery of bandlimited functions in higher dimensions

Abstract

We provide sufficient conditions on a family of functions (φα) α∈A : R d → R for sampling of multivariate bandlimited functions at certain nonuniform sequences of points in R d . We consider interpolation of functions whose Fourier transform is supported in some small ball in R d at scattered points (x j ) j∈N such that the complex exponentials e −i x j ,· j∈N form a Riesz basis for the L 2 space of a convex body containing the ball. Recovery results as well as corresponding approximation orders in terms of the parameter α are obtained.2000 Mathematics Subject Classification. Primary 41A05, 41A30, Secondary 42C30.

Compressing bistatic SAR target signatures with sparse‐limited persistence scattering models

“…We further assume that the functions described in Section 2.1 are sufficiently smooth and have a nearly band‐limited frequency spectrum. Using the assumptions stated above along with the fact that these functions have a limited support in and domain owing to the limited persistence, we hypothesise that these functions have a sparse representation in the set of multivariate Gaussian functions parametrised by the centre and the width of the Gaussian function by using the results presented in [22, 23] for function approximation and interpolation using Gaussian kernels. We utilise the following multivariate Gaussian functions in the bistatic angle and the pose denoted by with a structure as shown below where , , and are the centre and width of the Gaussian function in and domain, respectively.…”

Stability and Robustness of RBF Interpolation

“…For a multivariate analogue of Theorem 2.5, see [10,Theorem 3.6]; since its statement is somewhat technical and would take us out of the scope considered here, we omit it.…”

“…There are some known convergence phenomena and approximation results for bandlimited and Sobolev interpolation in the vein above. 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.…”

“…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.…”

On a realization of motion and similarity group equivalence classes of labeled points in $\mathbb R^k$ with applications to computer vision

“…We refer the reader to the following references which give a good perspective of Section (1.1) in various ways. [2,1,3,5,6,7,8,9,10,11,12,13,14,16,15].…”