Cardinal Interpolation With General Multiquadrics: Convergence Rates

TL;DR

This paper analyzes the convergence rates of cardinal interpolation using general multiquadrics for Sobolev functions, expanding known results and demonstrating optimal approximation rates in $L_p$ spaces.

Contribution

It extends the range of parameters for which convergence rates are known and connects cardinal interpolation with best approximation rates for multiquadrics.

Findings

Expanded convergence rate range for multiquadric-based interpolation.

Established $L_p$ error estimates in terms of lattice spacing $h$.

Showed that cardinal interpolants achieve known best approximation rates.

Abstract

This article pertains to interpolation of Sobolev functions at shrinking lattices $h\mathbb{Z}^d$ from $L_p$ shift-invariant spaces associated with cardinal functions related to general multiquadrics, $\phi_{\alpha,c}(x):=(|x|^2+c^2)^\alpha$. The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, $L_p$ error estimates in terms of the dilation $h$ are considered for the associated cardinal interpolation scheme. This analysis expands the range of $\alpha$ values which were previously known to give such convergence rates (i.e. $O(h^k)$ for functions with derivatives of order up to $k$ in $L_p$, $1<p<\infty$). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.