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Random sampling and polynomial-free interpolation by Generalized MultiQuadrics

A. Sommariva, M. Vianello

TL;DR

This work extends polynomial-free unisolvence results to Generalized MultiQuadrics (GMQ) in scattered interpolation by proving that the interpolation matrix $V_n=[ ext{φ}_j( extbf{x}_i)]$, where $ ext{φ}_j( extbf{x})= ext{φ}( orm{ extbf{x}- extbf{x}_j}_2)$ and $ ext{φ}(r)=(1+(ε r)^{2k})^{β}$ with order $ ext{⌈}β ext{⌉}>1$, is almost surely nonsingular under random sampling in any dimension. The proof embeds the determinant into a complex-analytic framework, writing $f( extbf{x})= ext{det}(A( extbf{x}))=- ext{det}(V_{n-1}) ext{φ}_n( extbf{x})^2+a( extbf{x}) ext{φ}_n( extbf{x})+b( extbf{x})$, and exploiting a branch point at $z_*= rac{1}{ε}e^{i rac{π}{2k}}$ to show $f$ cannot vanish identically. Consequently, the zero set of a nonzero analytic function has Lebesgue measure zero, yielding $ ext{prob}( ext{det}(V_{n+1})=0)=0$ and completing the induction. The result broadens polynomial-free unisolvence to the GMQ family and encompasses Buhmann–Ortmann generalizations, enabling stable, polynomial-free interpolation in multivariate settings without polynomial augmentation.

Abstract

We prove that interpolation matrices for Generalized MultiQuadrics (GMQ) of order greater than one are almost surely nonsingular without polynomial addition, in any dimension and with any continuous random distribution of sampling points. We also include a new class of generalized MultiQuadrics recently proposed by Buhmann and Ortmann.

Random sampling and polynomial-free interpolation by Generalized MultiQuadrics

TL;DR

This work extends polynomial-free unisolvence results to Generalized MultiQuadrics (GMQ) in scattered interpolation by proving that the interpolation matrix , where and with order , is almost surely nonsingular under random sampling in any dimension. The proof embeds the determinant into a complex-analytic framework, writing , and exploiting a branch point at to show cannot vanish identically. Consequently, the zero set of a nonzero analytic function has Lebesgue measure zero, yielding and completing the induction. The result broadens polynomial-free unisolvence to the GMQ family and encompasses Buhmann–Ortmann generalizations, enabling stable, polynomial-free interpolation in multivariate settings without polynomial augmentation.

Abstract

We prove that interpolation matrices for Generalized MultiQuadrics (GMQ) of order greater than one are almost surely nonsingular without polynomial addition, in any dimension and with any continuous random distribution of sampling points. We also include a new class of generalized MultiQuadrics recently proposed by Buhmann and Ortmann.
Paper Structure (2 sections, 1 theorem, 22 equations)

This paper contains 2 sections, 1 theorem, 22 equations.

Table of Contents

  1. Introduction
  2. Polynomial-free random GMQ interpolation

Key Result

Theorem 1

Let $\Omega$ be an open connected subset of $\mathbb{R}^d$, $d\geq 1$, and $\{\mathbf{x}_i\}_{i\geq 1}$ be a randomly distributed sequence on $\Omega$ with respect to any given probability density $\sigma(\mathbf{x})$, i.e. a point sequence produced by sampling a sequence of absolutely continuous ra be the interpolation matrix with respect to Generalized MultiQuadrics (GMQ) of order $\lceil\beta\r

Theorems & Definitions (1)

  • Theorem 1