Constrained approximation by splines with free knots

In this paper, a method that combines shape preservation and least squares approximation by splines with free knots is developed. Besides the coefficients of the spline a subset of the knot sequence, the so-calledfree knots, is included in the optimization process resulting in a nonlinear least squares problem in both the coefficients and the knots. The original problem, a special case of aconstrained semi-linear least squares problem, is reduced to a problem that has only the knots of the spline as variables. The reduced problem is solved by a generalized Gauss-Newton method. Special emphasise is given to the efficient computation of the residual function and its Jacobian. Dedicated to our colleague and teacher Prof. Dr. J. W. Schmidt on the occasion of his 65th birthday Research of the first author was supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1,2-2.     

Cubic Spline Prewavelets on the Four-Directional Mesh

Dedicated to Professor M. J. D. Powell on the occasion of his sixty-fifth birthday and his retirement. In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L ² (R ² ) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree.     

A note on the optimal stability of bases of univariate functions

This note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes.     

Radial basis functions under tension

We discuss multivariate interpolation with some radial basis function, called radial basis function under tension (RBFT). The RBFT depends on a positive parameter which provides a convenient way of controlling the behavior of the interpolating surface. We show that our RBFT is conditionally positive definite of order at least one and give a construction of the native space, namely a semi-Hilbert space with a semi-norm, minimized by such an interpolant. Error estimates are given in terms of this semi-norm and numerical examples illustrate the behavior of interpolating surfaces.     

Error estimates in Sobolev spaces for interpolating thin plate splines under tension

This paper discusses _Lp$L_p$-error estimates for interpolation by thin plate spline under tension of a function in the classical Sobolev space on an open bounded set with a Lipschitz-continuous boundary. A property of convergence is also given when the set of interpolating points becomes more and more dense.     

On the construction of local quadratic spline quasi-interpolants on bounded rectangular domains

In this paper local bivariate C¹$C^1$ spline quasi-interpolants on a criss-cross triangulation of bounded rectangular domains are considered and a computational procedure for their construction is proposed. Numerical and graphical tests are provided.     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

A shape preserving area true approximation of histogram by rational splines

For a given histogram, we consider an application of a simple rational spline to a shape preserving area true approximation of the histogram. An algorithm for determination of the spline is as easy as one with a quadratic polynomial spline, while the latter does not always preserve the shape of the histogram. Some numerical examples are given at the end of the paper.     

Fair upper bounds for the curvature in univariate convex interpolation

In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC ¹, cubicC ², and quarticC ³ splines on refined grids.     

Convex interval interpolation with cubic splines

A necessary and sufficient criterion is presented under which the problem of the convex interval interpolation with cubicC ¹-splines has at least one solution. The criterion is given as an algorithm which turns out to be effective.Dedicated to Professor Julius Albrecht on the occasion of his 60th birthday.     

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on Ω-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.     

On the optimal stability of bases of univariate functions

Summary. This paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation. The only bases considered are those whose elements have no sign changes. Among these, an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval. A uniqueness result and many examples of such optimally stable bases are also provided. Received May 26, 2000 / Published online August 17, 2001     

S-convex,monotone, and positive interpolation with rational bicubic splines of C2-continuity

In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC ² which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure.     

Thinning algorithms for scattered data interpolation

Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in ℝ ^d .     

Error estimates in W 2,∞-semi-norms for discrete interpolating D 2-splines

We derive error estimates in W^2,∞-semi-norms for multivariate discrete D²-splines that interpolate an unknown function at the vertices of given triangulations. These results are widely based on the construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local bases.     

Approximate Interpolation with Applications to Selecting Smoothing Parameters

In this paper, we study the global behavior of a function that is known to be small at a given discrete data set. Such a function might be interpreted as the error function between an unknown function and a given approximant. We will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region. We will apply these results to spline smoothing and show that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. This has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels.