Open Access
2017 Fractional cone and hex splines
Peter R. Massopust, Patrick J. Van Fleet
Rocky Mountain J. Math. 47(5): 1655-1691 (2017). DOI: 10.1216/RMJ-2017-47-5-1655


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.


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Peter R. Massopust. Patrick J. Van Fleet. "Fractional cone and hex splines." Rocky Mountain J. Math. 47 (5) 1655 - 1691, 2017.


Published: 2017
First available in Project Euclid: 22 September 2017

zbMATH: 1375.41005
MathSciNet: MR3705767
Digital Object Identifier: 10.1216/RMJ-2017-47-5-1655

Primary: 41A15 , 42A38 , 65D07

Keywords: $s$-dimensional mesh , (fractional) difference operator , box splines , Cone splines , fractional and complex B-splines , hex splines

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

Vol.47 • No. 5 • 2017
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