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Polynomial-free unisolvence of polyharmonic splines with odd exponent by random sampling

Alvise Sommariva, Marco Vianello

TL;DR

Problem: establish almost sure polynomial-free unisolvence for polyharmonic splines with odd exponent under random sampling, i.e., the interpolation matrix $V_n=[\phi(\|x_i-x_j\|_2)]$ with $\phi(r)=r^\nu$ and $\nu=2k+1$, remains nonsingular for all $n\ge 2$.\nApproach: treat the determinant as a real-analytic function of the centers, show the centers are distinct almost surely, and use an induction on $n$ together with the fact that the zero set of a nonzero analytic function on a connected domain has measure zero.\nContribution: proves almost-sure nonsingularity of $V_n$ for all $n\ge 2$, filling a gap left by prior work on odd exponent radial powers.\nImpact: provides a theoretical foundation for polynomial-free RBF interpolation for RP with odd exponents under random sampling, with implications for meshfree scattered-data methods.

Abstract

In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd exponents. In this short note we prove almost sure polynomial-free unisolvence in such instances, by a deeper analysis of the interpolation matrix determinant and fundamental properties of analytic functions.

Polynomial-free unisolvence of polyharmonic splines with odd exponent by random sampling

TL;DR

Problem: establish almost sure polynomial-free unisolvence for polyharmonic splines with odd exponent under random sampling, i.e., the interpolation matrix with and , remains nonsingular for all .\nApproach: treat the determinant as a real-analytic function of the centers, show the centers are distinct almost surely, and use an induction on together with the fact that the zero set of a nonzero analytic function on a connected domain has measure zero.\nContribution: proves almost-sure nonsingularity of for all , filling a gap left by prior work on odd exponent radial powers.\nImpact: provides a theoretical foundation for polynomial-free RBF interpolation for RP with odd exponents under random sampling, with implications for meshfree scattered-data methods.

Abstract

In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd exponents. In this short note we prove almost sure polynomial-free unisolvence in such instances, by a deeper analysis of the interpolation matrix determinant and fundamental properties of analytic functions.
Paper Structure (1 section, 1 theorem, 13 equations)

This paper contains 1 section, 1 theorem, 13 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1

Let $\Omega$ be an open connected subset of $\mathbb{R}^d$, $d\geq 2$, and $\{x_i\}_{i\geq 1}$ be a randomly distributed sequence on $\Omega$ with respect to any given probability density $\sigma(x)$, i.e. a point sequence produced by sampling a sequence of absolutely continuous random variables $\{

Theorems & Definitions (1)

  • Theorem 1