Chebyshev Splines Research Articles

Collocation by singular splines

Splines determined by the kernel of the differential operator \({D^{k}(D\sqrt{x}D)}\) are known to be useful to solve the singular boundary value problems of the form \({D\sqrt{x}Du=f(x,u)}\) . One of the most successful methods is the collocation method based on special Chebyshev splines. We investigate the construction of the associated B-splines based on knot-insertion algorithms for their evaluation, and their application in collocation at generalized Gaussian points. Specially, we show how to obtain these points as eigenvalues of a symmetric tridiagonal matrix of order k.

A few remarks on recurrence relations for geometrically continuous piecewise Chebyshevian B-splines

This works complements a recent article (Mazure, J. Comp. Appl. Math. 219(2):457–470, 2008) in which we showed that T. Lyche’s recurrence relations for Chebyshevian B-splines (Lyche, Constr. Approx. 1:155–178, 1985) naturally emerged from blossoms and their properties via de Boor type algorithms. Based on Chebyshevian divided differences, T. Lyche’s approach concerned splines with all sections in the same Chebyshev space and with ordinary connections at the knots. Here, we consider geometrically continuous piecewise Chebyshevian splines, namely, splines with sections in different Chebyshev spaces, and with geometric connections at the knots. In this general framework, we proved in (Mazure, Constr. Approx. 20:603–624, 2004) that existence of B-spline bases could not be separated from existence of blossoms. Actually, the present paper enhances the powerfulness of blossoms in which not only B-splines are inherent, but also their recurrence relations. We compare this fact with the work by G. Muhlbach and Y. Tang (Muhlbach and Tang, Num. Alg. 41:35–78, 2006) who obtained the same recurrence relations via generalised Chebyshevian divided differences, but only under some total positivity assumption on the connexion matrices. We illustrate this comparison with splines with four-dimensional sections. The general situation addressed here also enhances the differences of behaviour between B-splines and the functions of smaller and smaller supports involved in the recurrence relations.

On Chebyshevian spline subdivision

The paper addresses the question of convergence of Chebyshevian spline subdivision algorithms. This study suggests some ideas about the general treatment of non-interpolatory irregular (non-uniform/non-stationary) subdivision schemes.

B-Splines with Arbitrary Connection Matrices

We consider a space of Chebyshev splines whose left and right derivatives satisfy linear constraints that are given by arbitrary nonsingular connection matrices. We show that for almost all knot sequences such spline spaces have basis functions whose support is equal to the support of the ordinary B-splines with the same knots. Consequently, there are knot insertion and evaluation algorithms analogous to de Boor's algorithm for ordinary splines.

On nonlinear simultaneous Chebyshev approximation problems

This paper is concerned with the problem of nonlinear simultaneous Chebyshev approximation in a real continuous function space. Some results on existence are established, in addition to characterization conditions of Kolmogorov type and also of alternation type. Applications are given to approximation by rational functions, by exponential sums and by Chebyshev splines with free knots.

Quasi-Chebyshev splines with connection matrices: application to variable degree polynomial splines

Extending a result recently obtained for Chebyshevian splines, we give a necessary and sufficient condition for the existence of blossoms (or, equivalently, of B-spline bases) for splines with connection matrices and with sections in different four-dimensional quasi-Chebyshev spaces. We apply this result to the study of variable degree polynomial splines.

Chebyshev splines beyond total positivity

For polynomial splines as well as for Chebyshev splines, it is known that total positivity of the connection matrices is sufficient to obtain B-spline bases. In this paper we give a necessary and sufficient condition for the existence of B-bases (or, equivalently, of blossoms) for splines with connection matrices and with sections in different four-dimensional extended Chebyshev spaces.

Piecewise Smooth Spaces in Duality: Application to Blossoming

Given an (n+1)-dimensional space U of piecewise smooth functions in which each basis has a non-vanishing Wronskian, and its dual space U*, a canonical bilinear form is defined on U×U*, which provides a simple characterization of a contact of order r⩽n. An intrinsic reproducing function is introduced, leading to Marsden-type identities. In the case of Chebyshev spaces connected with totally positive matrices, the bilinear form yields a general notion of blossom which can be extended to Chebyshev splines.

Polynomial Chebyshev splines

A particular type of Chebyshev spaces included in polynomial spaces is studied leading to new shape parameters. Chebyshev splines based on these spaces are defined and the corresponding shape effects are analyzed.

GB-splines of arbitrary order

Explicit formulae and recurrence relations for the calculation of generalized B-splines (GB-splines) of arbitrary order are given. We derive main properties of GB-splines and their series, i.e. partition of unity, shape-preserving properties, invariance with respect to affine transformations, etc. It is shown that such splines have the variation diminishing property and are Chebyshevian splines.

E-optimal designs for polynomial spline regression

We give the E-optimal approximate designs for mean (sub-) parameters in dth degree totally positive polynomial spline regression with prescribed knots over an arbitrary compact real interval. Based on a duality between E- and scalar optimality, the optimal design is found to be supported by the extrema of the Chebyshev (i.e., equi-oscillating) spline, with corresponding weights given in terms of certain Lagrange interpolation splines. In particular, for dth degree polynomial regression, parameterized w.r.t. a totally positive basis (e.g. the Bernstein polynomials), we obtain the solution to the E-optimal design problem, where, contrary to the ordinary monomial setup, no restrictions on the size and location of the regression interval or on the particular system of parameters of interest are required.

On the approximation by Chebyshev splines in the metric of L p , p > 0

We prove a direct Jackson estimate for the approximation by Chebyshev splines in the classes Lp, p > 0.

Approximation of certain classes of differentiable functions by generalized splines

We find the exnet value of the best (α, β)-approximation by generalized Chebyshev splines for a class of functions differentiable with weight on [−1, 1].

Basis of splines associated with some singular differential operators

Functions being piecewise in Ker (D k DpD) are a special case of Chebyshev splines having one nontrivial weight and also a special case of singular splines. An algorithm is designed which enables calculating with related B-splines and their derivatives. Ifp(t) is approximated by a piecewise constant, an interesting recurrence for calculating with polynomial B-splines is obtained.

Birkhoff interpolation by Chebyshevian splines

This paper deals with the interpolation of the function and its derivative values at scatted points, so-called Birkhoff Interpolation, by piecewise Chebyshevian spline.

Best L1-approximation by generalized convex functions

Let {u 0,…, u n − 1} be an ECT-system and let C( u 0,…, u n − 1 ) be its generalized convexity cone. We give a precise description of best L 1-approximations to continuous functions from C( u 0,…, u n − 1 ). These best approximations are, piecewise, certain extremal Chebyshevian splines, which are obtained by applying results from moment theory for Chebyshev systems.

Collocation by L-Splines at Transformed Gaussian Points

A class of L-splines is used to solve two-point boundary value problems. This class includes the set of Chebyshevian splines. The set of collocation points is a member of a general family related to the Gaussian points. We achieve the same order of convergence as for polynomial splines collocating at Gaussian points. A formula for the leading part of the error is found. An application is made to trigonometric splines. We show that these are useful in solving periodic boundary value problems.

Generalized Chebyshevian Splines

In this paper we study a space of splines in which the pieces are drawn from a linear space spanned by an ECT-system U. The splines here generalize the usual Chebyshevian splines in that the pieces in the various intervals are restricted to come from varying subspaces of U. For our class of generalized splines we discuss zeros, determinants associated with certain Lagrange and Hermite interpolation problems, and properties of certain local support basis splines.

Weak Chebyshev sets and splines

Weak Chebyshev sets and splines

On uniqueness of best spline approximations with free knots

This paper is concerned with Chebyshev approximation by spline functions with free knots. If a zero of a Chebyshev spline function occurs at a knot, the multiplicity of the zero is suitably extended. Theorems on uniqueness on the whole approximation interval and on subintervals are stated in terms of alternation properties.