## Rocky Mountain Journal of Mathematics

### 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.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 5 (2017), 1655-1691.

Dates
First available in Project Euclid: 22 September 2017

https://projecteuclid.org/euclid.rmjm/1506045627

Digital Object Identifier
doi:10.1216/RMJ-2017-47-5-1655

Mathematical Reviews number (MathSciNet)
MR3705767

Zentralblatt MATH identifier
1375.41005

#### Citation

Massopust, Peter R.; Fleet, Patrick J. Van. Fractional cone and hex splines. Rocky Mountain J. Math. 47 (2017), no. 5, 1655--1691. doi:10.1216/RMJ-2017-47-5-1655. https://projecteuclid.org/euclid.rmjm/1506045627

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