Recently, we have introduced spaces of splines deened on triangulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein-B ezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangula-tions. ...

A basic property of both polynomial and more general Tchebycheffian splines is that the associated B-spline collocation matrix has certain total positivity properties, and it is nonsingular if and only if well-known interlacing conditions hold. Using a technique from a recent paper of Mørken, we extend these results to L-splines. §

We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r ≥ 1 and polynomial degree 3r − 1. They are defined on a triangulation with PowellSabin refinement. The basis functions have a local support, they are nonnegative and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set ...

We establish the asymptotical equivalence between L-spline smoothing and kernel estimation. The equivalent kernel is used to derive the asymptotic mean squared error of the L-smoothing spline estimator. The paper extends the corresponding results for polynomial spline smoothing.

A method of generating cubic blending spline curves based on weighted trigonometric and hyperbolic polynomial is presented in this paper. The curves inherit nearly all properties of cubic B-splines and enjoy some other advantageous properties for modeling. They can represent some conics and some transcendental curves exactly. Here weight coefficients are also shape parameters, which are called ...

In science and engineering there often is a need for the approximation of scattered multi-dimensional data. A class of powerful scattered data approximators are the multivariate simplex B-splines. Multivariate simplex B-splines consist of Bernstein basis polynomials that are defined on a geometrical structure called a triangulation. Multivariate simplex B-splines have a number of advantages ove...

Differential operators and associated boundary conditions have been used to define interpolating splines, such as L-splines, since the mid 1960s. This work adapts that approach to define kernels which can be used to perform equivalent interpolation. While a closed form of these kernels is generally unavailable, their eigenexpansions can be determined by solving the Sturm-Liouville eigenvalue pr...

The purpose of this paper is to discuss numerical solutions of differential equations including the evolution, progress and types of differential equations. Special attention is given to the solution of differential equations by application of spline functions. Here we are interested in differential equation based problems and their solutions using polynomial and nonpolynomial splines of differ...

A flexible nonparametric regression model is considered in which the response depends linearly on some covariates, with regression coefficients as additive functions of other covariates. Polynomial spline estimators are proposed for the unknown coefficient functions, with optimal univariate mean square convergence rate under geometric mixing condition. Consistent model selection method is also ...