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

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.