Complex homogeneous splines on the torus

Abstract

We construct functions M_α which are piecewise homogeneous polynomials on the (d+1)-dimensional torus U^d+1. These functions possess complete symmetry with respect to the independent variables. The symmetry and homogeneous relations for these functions are exploited to obtain a recurrence relation and explicit representations. Furthermore, we show that$M_\alpha \left( {z_0 ,\omega ^{ - j_1 } z_1 , \cdots ,\omega ^{ - i_d } z_d } \right), \left( {z_0 , \cdots ,z_d } \right) \in U^{d + 1}$, where ω=e^12x/k, 0≤j_t≤k−1, are linearly independent. By restricting M_α to U^d, we obtain the complex analogue of polynomial box splines on a (d+1)-direction mesh on U^d, which is a multivariate analogue of B-splines on the circle studied by I.J. Schoenberg^[8].