Construction of Bases in Modules over Laurent Polynomial Rings and Applications to Box Spline Prewavelets

TL;DR

This paper introduces a novel basis construction method for modules over Laurent polynomial rings, enhancing multivariate wavelet applications and producing more efficient box spline prewavelets with smaller supports and fewer coefficients.

Contribution

It presents a new basis construction technique for Laurent polynomial modules, improving the design of multivariate wavelets and box spline prewavelets with optimized support and coefficient properties.

Findings

Smaller mask supports for $C^1$ cubic and $C^2$ quartic box splines in two variables.

Trivariate piecewise linear prewavelets with at most 23 nonzero mask coefficients.

Outperforms previous constructions in efficiency and support size.

Abstract

We suggest a new method of basis construction for the kernel of a linear form on the Laurent polynomial module related to multivariate wavelets, and demonstrate its applications to box spline prewavelets, leading to small mask supports for $C^1$ cubic and $C^2$ quartic box splines in two variables, outperforming previously known constructions, and to trivariate piecewise linear prewavelets with at most 23 nozero mask coefficients.