Module Bases for Multivariate Splines

Fractional Splines , Wavelet Bases

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

Multivariate Adaptive Regression Splines

Bases for Splines on a Subdivided Domain

Let Sr(∆) be the module of all splines of smoothness r on the rectilinear partition ∆ which subdivides some domain D. Further, let Sr(Γ) be the module of all splines of smoothness r on Γ which also subdivides D, where Γ is a finer subdivision of ∆. We study the relationship between a generating set of Sr(∆) and a generating set for Sr(Γ). This paper gives an algorithm for extending a generating...

Generating dimension formulas for multivariate splines

Dimensions of spaces of multivariate splines remain unknown in general. A computational method to obtain explicit formulas for the dimension of spline spaces on simplicial partitions is described. The method is based on Hilbert series and Hilbert polynomials. It is applied to conjecture the dimension formulas for splines on the Alfeld split of a simplex and on several other partitions. AMS clas...

Sensible parameters for univariate and multivariate splines

The package bspline, downloadable from SSC, now has 3 modules. The first, bspline, generates a basis of Schoenberg B-splines. The second, frencurv, generates a basis of reference splines, whose parameters in the regression model are simply values of the spline at reference points on the Xaxis. The recent addition, flexcurv, is an easy–to–use version of frencurv, and generates reference splines ...