Anisotropic spline approximation with non-uniform B-splines

TL;DR

This paper introduces a new constructive approach to non-uniform B-splines using diversification, enabling effective anisotropic approximation of functions on complex domains with shape-independent error bounds.

Contribution

It presents a novel modification of non-uniform B-splines based on diversification, along with a bounded quasi-interpolant and shape-independent anisotropic error estimates.

Findings

Error constants are independent of domain shape.

The method provides effective anisotropic approximation in $L^p$-norm.

Spline spaces are well-suited for functions on irregular domains.

Abstract

Recently the author and U. Reif introduced the concept of diversification of uniform tensor product B-splines. Based on this concept, we give a new constructive modification of non-uniform B-splines. The resulting spline spaces are perfectly fitted for the approximation of functions defined on domains $\Omega\subset \mathbb{R}^2$. We build a bounded quasi-interpolant and prove that for our spline spaces an anisotropic error estimate in the $L^p$-norm, $1\le p\le\infty$, is valid. In particular, we show that the constant of the error estimate does not depend on the shape of $\Omega$ or the knot grid.