End conditions for improved cubic spline derivative approximations

We consider the problem of deriving accurate end conditions for cubic spline interpolation at equally spaced knots. In particular we derive a number of end conditions which lead to derivative approximations of high accuracy.     

End Conditions for Interpolatory Quintic Splines

Present address: Department of Mathematics, University of Tabriz, Tabriz, Iran. Accurate end conditions are derived for quintic spline interpolationat equally spaced knots. These conditions are in terms of availablefunction values at the knots and lead to O(h⁶) convergence uniformlyon the interval of interpolation.     

Interpolation and Approximation by Splines on [0, ∞ )

We investigate interpolation and approximation problems by splines, which possess a countable set of knots on the positive axis. In particular, we characterize those sets of points, which admit unique Lagrange interpolation and give some sufficient and some necessary conditions for best approximations. Moreover, we show that the classical results of spline-approximation theory are not available for splines with a countable set of knots.     

О наилучшем выборе уз лов при интерполиров ании функций эрмитовыми с плайнами

The problem of optimal choice of knots is considered for the functions belonging to the classW ^2m+1 V, concerning interpolation by means of Hermite splines. The problem of asymptotically best choice of the knots for interpolation of a fixed functionf(x) (f^(2m+2)(x)>0, 0x1) by Hermite splines is also treated.     

Interpolation by splines satisfying mixed boundary conditions

We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen?k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.     

Cubic Splines with Minimal Norm

Natural cubic interpolatory splines are known to have a minimal L ₂-norm of its second derivative on the C ² (or W ₂ ² ) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C ¹ splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.     

An always successful method in univariate convex C^2 interpolation

Summary. We consider convex interpolation with cubic splines on grids built by adding two knots in each subinterval of neighbouring data sites. The additional knots have to be variable in order to get a chance to always retain convexity. By means of the staircase algorithm we provide computable intervals for the added knots such that all knots from these intervals allow convexity preserving spline interpolation of continuity. Received May 31, 1994 / Revised version received December 22, 1994     

Shape-preserving interpolation by splines using vector subdivision

We give a local convexity preserving interpolation scheme using parametricC ² cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.     

On integro quartic spline interpolation

In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others.     

On Optimal Error Bounds for Derivatives of Interpolating Splines on a Uniform Partition

Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros.     

Interpolation by periodic splines with Birkhoff knots

Summary We give a complete characterization of the Hermite interpolation problem by periodic splines with Birkhoff knots. As a dual result we derive the characterization of the Birkhoff interpolation by periodic splines with multiple knots.Sponsored by the Bulgarian Ministry of Education and Science under Contract No. MM-15     

Cubic spline interpolation with quasiminimal B-spline coefficients

Summary The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.     

Bounds on the solutions of difference equations and spline interpolation at knots

We prove several comparison theorems for difference equations and discuss their application to spline interpolation at knots.     

样条最佳插值结点的非线性规划算法

§1.问题的提出 [1]研究了二阶算子样条最佳插值结点的特征.对于少数几个函数,利用特征定理精确求出了其最佳插值结点.但是,如[1]中指出,对于绝大多数函数,要精确求出其最佳插值结点,是相当困难的.因此,设计相应的数值求解方法,对于实际应用是很有必要的.     

Discrete cubic spline interpolation

Summary In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.     

具有重结点的B-样条曲面G1的连续条件

本文得到了关于两个双三次内部重结点B-样条曲面片G¹连续的充分必要条件和在公共边界线上控制向量的本征条件.这些条件直接由两个B-样条曲面的控制向量表示.文[10]证明了使用内部单结点的双三次B-样条曲面来构造G¹光滑曲面,局部格式不存在.使用本文的这些条件就可以构造出具有局部各式的G¹光滑造型.     

The Budan-Fourier theorem for splines

The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the number of sign changes in the sequence of successive derivatives evaluated at the end-points. An extension is offered to splines with knots of arbitrary multiplicities, in which case the connection involves the number of zeros of the highest derivative. The theorem yields bounds on the number of zeros of splines and is a valuable tool in spline interpolation and approximation with boundary conditions.     

On the Construction of Optimal Monotone Cubic Spline Interpolations

In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs.     

Nonnegative splines, in particular of degree five

Summary. We investigate splines from a variational point of view, which have the following properties: (a) they interpolate given data, (b) they stay nonnegative, when the data are positive, (c) for a given integer they minimize the functional for all nonnegative, interpolating . We extend known results for to larger , in particular to and we find general necessary conditions for solutions of this restricted minimization problem. These conditions imply that solutions are splines in an augmented grid. In addition, we find that the solutions are in and consist of piecewise polynomials in with respect to the augmented grid. We find that for general, odd there will be no boundary arcs which means (nontrivial) subintervals in which the spline is identically zero. We show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval. For we show that between given neighboring interpolation knots, the augmented grid has at most two additional grid points. In the case of two interpolation knots (the local problem) we develop polynomial equations for the additional grid points which can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. There are extensions to other types of constraints such as two-sided restrictions, also ones which vary from interval to interval. As an illustration several numerical examples including graphs of splines manufactured by MATLAB- and FORTRAN-programs are given. Received November 16, 1995 / Revised version received February 24, 1997     

Spline interpolation at knot averages

It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork 20.Communicated by Wolfgang Dahmen.