Left Inverses for B-spline Subdivision Matrices in Tensor-Product Spaces

TL;DR

This paper investigates local dyadic coarsening operators in univariate and tensor-product spline spaces, providing efficient approximations comparable to global best fits, inspired by prior work on local least squares.

Contribution

It introduces new local inverse operators for B-spline subdivision matrices in tensor-product spaces, enhancing computational efficiency over existing global methods.

Findings

Operators are local and computationally inexpensive.

Approximations are comparable to global L2-best.

Applicable to univariate and tensor-product spline spaces.

Abstract

In this article, we study dyadic coarsening operators in univariate spline spaces and in tensor-product spline spaces over uniform grids. Our construction is strongly motivated by the work of Bartels, Golub, and Samavati (2006), Some observations on local least squares, BIT, 46(3):455--477. The proposed operators are local in nature and yield approximations to a given spline that are comparable to the global L2-best approximation, while being significantly faster to compute and computationally inexpensive.