Near-Best Univariate Spline Discrete Quasi-Interpolants on Nonuniform Partitions

The univariate spline quasi-interpolants (QIs) studied in this paper are approximation operators using B-spline expansions with coefficients that are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs that have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound for the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of some families of near-best discrete quasi-interpolants (dQIs) of approximation order 3.      

A Family of Spline Quasi-Interpolants on the Sphere

In this paper, we construct a local quasi-interpolant Q for fitting a function f defined on the sphere S. We first map the surface S onto a rectangular domain and next, by using the tensor product of polynomial splines and 2-periodic trigonometric splines, we give the expression of Qf. The use of trigonometric splines is necessary to enforce some boundary conditions which are useful to ensure the C ² continuity of the associated surface. Finally, we prove that Q realizes an accuracy of optimal order.     

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane

In this paper we present a study of spaces of splines in C ^k (R ²) with supports the square ₁ and the lozenge ₁ formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called ₁ and ₁-splines. We first compute the dimension of the space of ₁-splines. Then we prove the existence of a unique ₁-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique ₁-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of ₁-spline and ₁-spline of minimal degree.