Periodic exponential box splines on a three direction mesh

Abstract

Multivariate “truncated Tchebycheff” functions which generalize the univariate Green's function on the one hand and the multivariate “truncated powers” on the other hand are constructed and analysed for the three direction mesh. For weight functions which are products of an exponential and a periodic function, the application of suitable divided differences to the corresponding truncated Tchebycheff functions yields a new type of box spline termed “periodic exponential.” This class of box splines contains the exponential (hence polynomial) box splines of the three direction mesh as special cases. It also extends the notion of the univariate periodic exponential Tchebycheffian B-splines.