Spline smoothing under constraints on derivatives

An efficient algorithm for computing a smoothing polynomial splines under inequality constraints on derivatives is introduced where both order and breakpoints ofs can be prescribed arbitrarily. By using the B-spline representation ofs, the original semi-infinite constraints are replaced by stronger finite ones, leading to a least squares problem with linear inequality constraints. Then these constraints are transformed into simple box constraints by an appropriate substitution of variables so that efficient standard techniques for solving such problems can be applied. Moreover, the smoothing term commonly used is replaced by a cheaply computable approximation. All matrix transformations are realized by numerically stable Givens rotations, and the band structure of the problem is exploited as far as possible.     

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.     

Piecewise linear prewavelets on arbitrary triangulations

This paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology. It is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high. Received December 29, 1997 / Revised version received April 14, 1998     

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.     

On consistency relations for polynomial splines on a uniform partition

In this paper we present linear dependence relations connecting spline values, derivative values and integral values of the spline. These relations are useful when spline interpolants or histospline projections of a function are considered.This work was supported in part by the Ministère de l'Éducation du Québec and by the Department of the National Defence of Canada.     

Spline approximant in Hilbertian subspaces ofC(Ω)

Summary In this paper we give a new approach of approximation by spline functions. We define and study approximant spline functions which can be easly calculated without solving a linear system. We investigate also the error in using approximant spline functions.     

From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms

In this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension.     

Optimization of parameters for curve interpolation by cubic splines

In this paper, we present an interpolation method for curves from a data set by means of the optimization of the parameters of a quadratic functional in a space of parametric cubic spline functions. The existence and the uniqueness of this problem are shown. Moreover, a convergence result of the method is established in order to justify the method presented. The aforementioned functional involves some real non-negative parameters; the optimal parametric curve is obtained by the suitable optimization of these parameters. Finally, we analyze some numerical and graphical examples in order to show the efficiency of our method.     

Approximation by smoothing variational vector splines for noisy data

This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of D^m${\mathrm{D}}^m$-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given.     

Elementarym-harmonic cardinal B-splines

We generalize the notion of B-spline to the thin plate splines and to otherd-dimensional polyharmonic splines as defined in [Duchon, [3]]; for regular nets, we give the main properties of these B-splines: Fourier transform, decay when x , stability, integration property, links between B-splines of different orders or of different dimensions and in particular link with the polynomial B-splines, approximation using B-splines... We show that, in some sense, B-splines may be considered as a regularized form of the Dirac distribution.     

Comonotone shape-preserving spline histopolation

Algorithms are presented for the construction of histopolating splines consisting of linear/linear rational or quadratic polynomial pieces. A unique comonotone histospline of such kind exists for any histogram with weak alternation of data. In general case, without weak alternation of data, a modified comonotone spline histopolation strategy should be used. The method is implemented via the representation with histogram heights and knot values of first derivatives of the spline. Numerical examples are given.     

High levelm-harmonic cardinal B-splines

We generalize the notion ofm-harmonic cardinal B-spline defined in [Rabut, [6c]] to obtain B-splines on an infinite regular grid, which are halfway between elementary B-splines and the Lagrangean cardinal spline function. We give the main properties of these functions: Fourier transform, decay when x , integration,P _k -reproduction (fork<-2m–1) of the associated B-spline approximation, etc. We show that, in some sense, high levelm-harmonic B-splines may be considered as a finer regular approximation of the Dirac distribution than the elementarym-harmonic B-splines are.     

Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines

Summary A method is presented for fitting a function defined on a general smooth spherelike surfaceS, given measurements on the function at a set of scattered points lying onS. The approximating surface is constructed by mapping the surface onto a rectangle, and using a tensor-product of polynomial splines with periodic trigonometric splines. The use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points onS. Two alternative algorithms for computing the coefficients of the tensor fit are presented; one based on global least-squares, and the other on the use of local quasi-interpolators. The approximation order of the method is established, and the numerical performance of the two algorithms is compared.Supported in part by the National Science Foundation under Grant DMS-8902331 and by the Alexander von Humboldt Foundation     

Low degree rational spline interpolation

For a strictly monotone functionf on [a,b] we describe the possibility of finding an interpolating rational splineS of the formS(x)=c ₀ +c ₁ x/(1+d ₁ x) on each subinterval of the grida=x ₀ <x ₁ <...<x _n =b. This leads to a nonlinear system for which we get the local existence and uniqueness of a solution. We prove that ‖S−f‖_∞=O(h ³). Numerical test shows good approximation properties of these splines.     

OnG 2 continuous cubic spline interpolation

In this paper the interpolation byG ² continuous planar cubic Bézier spline curves is studied. The interpolation is based upon the underlying curve points and the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case. Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory. Supported in prat by the Ministry of Science and Technology of Slovenjia, and in part by the NSF and SF of National Educational Committee of China.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Algorithms for spline wavelet packets on an interval

The aim of this paper is to describe decomposition and reconstruction algorithms for spline wavelet packets on a closed interval. In order to generate packet spaces of dyadic dimensions, it is necessary to modify the approach for spline wavelets on an interval as studied by Chui, Quak and Weyrich in [3, 11]. The first author was supported by the Department of the Air Force, contract F33600-94-M-2603, and the second author by the Department of Defense, contract H98230-R5-93-9187.     

B-splines with Birkhoff knots

A natural extension of the Curry-SchoenbergB-splines is given, which preserves such critical properties as variation diminishing and total positivity. Using this tool we give a characterization of the Birkhoff interpolation problem for spline functions.Communicated by Dietrich Braess.     

Cubature rule associated with a discrete blending sum of quadratic spline quasi-interpolants

A new cubature rule for a parallelepiped domain is defined by integrating a discrete blending sum of C¹ quadratic spline quasi-interpolants in one and two variables. We give the weights and the nodes of this cubature rule and we study the associated error estimates for smooth functions. We compare our method with cubature rules based on the tensor products of spline quadratures and classical composite Simpson’s rules.     

Two shape preserving lagrangeC 2-interpolants

Summary We present a LagrangeC ²-interpolant to scattered convex data which preserves convexity. We also present a LagrangeC ²-interpolant to uniformly spaced monotone data sites which preserves monotonicity. In both cases no further conditions are required on the data values. These interpolants are explicitely described and local. Error isO(h ³) when the function to be interpolated isC ³.