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arXiv · 1504.00546

Fractional Cone Splines and Hex Splines

Peter R. Massopust·Patrick J. Van Fleet
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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 $s$-dimensional meshes and include as special cases the three-directional box splines \cite{article:condat} and hex splines \cite{article:vandeville} previously considered by Condat, 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:ub,article:fbu} and investigated in, e.g., \cite{article:fm,article:mf}. Explicit time domain representations are derived for these splines on $3$-directional meshes. We present some properties of these two multivariate spline families such as recurrence, decay and refinement. Finally it is shown that a bivariate hex spline and its integer lattice translates form a Riesz basis of its linear span.

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Peter R. Massopust, Patrick J. Van Fleet. 2015-04-02. Fractional Cone Splines and Hex Splines. https://arxiv.org/abs/1504.00546

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