Convergence Properties of Spline-Like Cardinal Interpolation Operators Acting on $$\ell ^p$$ ℓ p Data

Abstract

In this article, we provide examples of interpolation and approximation methods for p data. We also show that the resulting interpolants share convergence properties similar to those enjoyed by splines.

Cardinal interpolation with general multiquadrics

“…Radial basis cardinal interpolation also enjoys a strong connection with classical sampling theory, as evidenced by much of the aforementioned literature. This connection continues to be explored in recent developments by the second author [23,24], and by parts of this article. As this is one of the most interesting aspects of cardinal interpolation, we provide some of the motivation.…”

Regular families of kernels for nonlinear approximation

“…In particular, we assume from here on that φ > 0 on R d . Cardinal functions associated with radial basis functions have been studied rather extensively [4,5,10,11,22,23,25,33,34,[36][37][38][39]41]. Specifically, the cardinal functions associated with the Gaussian and general multiquadrics are known to have nice decay (exponential in the former case and polynomial based on the exponent α in the latter).…”

“…. , c 7 ) 100 (10,16,16,9,28,20, 2) 200 (21,31,31,17,56,39,5) 300 (31,47,47,26,83,59,7) 400 (42,62,62,34,111, 79, 10) 500 (52,78,78,43,139,98,12) Table 1. Cost distribution for f with sparse wavelet representation.…”

On regular families of cardinal interpolators and multiresolution analyses

“…Some of this connection has been detailed in [2], where similar L 2 convergence properties are exhibited for the associated fundamental functions of interpolation. A result concerning general L p , 1 < p < ∞ results may be found in [3]. In [4], it was shown that polyharmonic splines may be used to generate multiresolution analyses.…”