Anisotropic spline approximation with non-uniform B-splines

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.