Quadratic spline wavelets with arbitrary simple knots on the sphere

AbstractIn this paper, we extend the method for fitting functions on the sphere, described in Lyche and Schumaker (SIAM J. Sci. Comput. 22 (2) (2000) 724) to the case of nonuniform knots. We present a multiresolution method leading to C1-functions on the sphere, which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the tensor product decomposition and reconstruction algorithms in matrix forms which are convenient for the compression of surfaces. We give the different steps of computer implementation and finally we present a test example by using two knot sequences: a uniform one and a sequence of Chebyshev points

Near-best quasi-interpolants associated with H-splines on a three-directional mesh.

International audienceSpline quasi-interpolants with best approximation orders and small norms are useful in several applications. In this paper, we construct the so-called near-best discrete and integral quasi-interpolants based on H-splines, i.e., B-splines with regular hexagonal supports on the uniform three-directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials, and minimize an upper bound of their infinite norms depending on a finite number of free parameters. We show that this problem has always a solution, but it is not unique in general. Concrete examples of these types of quasi-interpolants are given in the two last sections

Near-best bivariate spline quasi-interpolants on a four-directional mesh of the plane

Spline quasi-interpolants (QIs) are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete and integral quasi-interpolants which are based on $\Omega$-~splines, i.e. B-splines with regular lozenge supports on the uniform four directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinity norms which depend on a finite number of free parameters. We show that this problem has always a solution, which is not unique in general. Concrete examples of these types of quasi-interpolants are given in the last section

Near minimally normed spline quasi-interpolants on uniform partitions

International audienceSpline quasi-interpolants are local approximating operators for functions or discrete data. We consider the construction of discrete and integral spline quasi-interpolants on uniform partitions of the real line having small infinite norms. We call them near minimally normed quasi-interpolants: they are exact on polynomial spaces and minimize a simple upper bound of their infinite norms. We give precise results for cubic and quintic quasi-interpolants. Also the quasi-interpolation error is considered, as well as the advantage that these quasi-interpolants present when approximating functions with isolated discontinuities

Near-best univariate spline discrete quasi-interpolants on non-uniform partitions

International audienceUnivariate spline discrete quasi-interpolants (abbr. dQIs) are approximation operators using B-spline expansions with coefficients which are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find dQIs which have both good approximation orders and bounded uniform norms independent of the given partition. Near-best dQIs are obtained by minimizing an upper bound of the infinite norm of dQIs 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 dQIs of approximation order 2