Optimal Compactly Supported Functions in Sobolev Spaces

TL;DR

This paper constructs and analyzes optimal compactly supported functions in Sobolev spaces, providing tools for improved interpolation, approximation, and PDE methods with proven convergence rates.

Contribution

It introduces a method to construct unique, optimal compactly supported functions in Sobolev spaces with minimal norm and maximal support, useful for approximation and PDE solving.

Findings

Derived the optimal convergence rate $m-d/2$ for Sobolev space interpolation.

Constructed explicit functions with maximal support and minimal norm in Sobolev spaces.

Provided numerical examples demonstrating the theoretical results.

Abstract

This paper constructs unique compactly supported functions in Sobolev spaces that have minimal norm, maximal support, and maximal central value, under certain renormalizations. They may serve as optimized basis functions in interpolation or approximation, or as shape functions in meshless methods for PDE solving. Their norm is useful for proving upper bounds for convergence rates of interpolation in Sobolev spaces $H_2^m(\R^d)$, and this paper gives the correct rate $m-d/2$ that arises as convergence like $h^{m-d/2}$ for interpolation at meshwidth $h\to 0$ or a blow-up like $r^{-(m-d/2)}$ for norms of compactly supported functions with support radius $r\to 0$. In Hilbert spaces with infinitely smooth reproducing kernels, like Gaussians or inverse multiquadrics, there are no compactly supported functions at all, but in spaces with limited smoothness, compactly supported functions exist and can be optimized in the above way. The construction is described in Hilbert space via projections, and analytically via trace operators. Numerical examples are provided.