On the Number of Tight Wavelet Frame Generators associated with Multivariate Box Splines

We study the number of Laurent polynomials in a sum of square magnitudesof a nonnegative Laurent polynomial associated with bivariate box splines on the three-and the four-direction meshes. In addition, we study the same problem associated withtrivariate box splines. The number of Laurent polynomials in a sum of squares magnitudeof a nonnegative Laurent polynomial determines the number of multivariate box splinetight wavelet frame generators under the simple construction method in [5]. As a result,the numbers of bivariate box spline tight wavelet frame generators on the threeand thefour-direction meshes under certain condition and the one for trivariate box splines aremuch smaller than the number of multivariate box spline tight wavelet frame generatorsconstructed via Kronecker poduct method in [2].