Fractional cone and hex splines

Abstract

We introduce an extension of cone splines and box splines to fractional and complex orders. These new families of multivariate splines are defined in the Fourier domain along certain $s$-directional meshes and include as special cases the $3$-directional box splines~\cite {article:condat} and hex splines~\cite {article:vandeville} previously considered by Condat and Van De Ville, et al. These cone and hex splines of fractional and complex order generalize the univariate fractional and complex B-splines defined in~\cite {article:fbu, article:ub} and, e.g., investigated in~\cite {article:fm, article:mf}. Explicit time domain representations are de\-rived for these splines on $3$-directional meshes. We present some properties of these two multivariate spline families, such as recurrence, decay and refinement. Finally, we show that a bivariate hex spline and its integer lattice translates form a Riesz basis of its linear span.