-error estimates for ``shifted' surface spline interpolation on Sobolev space

The accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant is reasonably smooth. However, for functions which have smoothness below a certain order associated with the basis function , no approximation power has yet been established. Hence, the purpose of this study is to discuss the -approximation order ( ) of interpolation to functions in the Sobolev space with \max(0,d/2-d/p)$">. We are particularly interested in using the ``shifted' surface spline, which actually includes the cases of the multiquadric and the surface spline. Moreover, we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met.

     

The -approximation order of surface spline interpolation

We show that if the open, bounded domain has a sufficiently smooth boundary and if the data function is sufficiently smooth, then the -norm of the error between and its surface spline interpolant is ( ), where and is an integer parameter specifying the surface spline. In case , this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the -approximation order of surface spline interpolation is .

     

Cardinal Hermite interpolation using positive definite functions

In this paper, we study cardinal Hermite interpolation by using positive definite functions. Among other things, we establish a procedure that employs the multiquadrics for cardinal Hermite interpolation.     

紧支撑样条小波插值及其应用

基于紧支撑样条小波函数插值与定积分的思想,给出了由紧支撑样条小波插值函数构造数值积分公式的方法．并将该方法应用于二次、三次、四次和五次紧支撑样条小波函数,得到了相应的数值积分公式．最后,通过数值例子验证,发现该方法得到的数值积分公式是准确的,且具有较高精度．     

基样条与整函数的极限性质

本文证明了‖smf-f‖ p→ 0 ( m→∞ )的必要条件是 f∈ Bπ,p,其中 Bπ,p=Bπ∩ Lp( R) ,Bπ表示指数 π型的整函数在R上限制是有界函数所构成的集合 ,smf 是在整数点对 f 插值的唯一确定的 m-1次基样条 .最终得到了关于整函数的一个等价刻划     

Lebesgue constant for Lagrange interpolation on equidistant nodes

Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobt polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.     

Cardinal interpolation and spline fucntions V. The B-splines for cardinal Hermite interpolation

In the third paper of this series on cardinal spline interpolation [4] Lipow and Schoenberg study the problem of Hermite interpolation

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′,…,S}{}^{\mathrm{(r?1)}}\text{(v) = Y}{}_{\mathrm{v}}{}^{\mathrm{(r?1)}}\text{for all}\text{v}$

. The B-splines are there conspicuous by their absence, although they were found very useful for the case γ = 1 of ordinary (or Lagrange) interpolation (see [5–10]). The purpose of the present paper is to investigate the B-splines for the case of Hermite interpolation (γ > 1). In this sense the present paper is a supplement to [4] and is based on its results. This is done in Part I. Part II is devoted to the special case when we want to solve the problem

$\text{S(v) = Y}{}_{\mathrm{v}}\text{, S′(v) = Y}{}_{\mathrm{v}}\text{′}\text{for all}\text{v}$

by quintic spline functions of the class C?(– ∞, ∞). This is the simplest nontrivial example for the general theory. In Part II we derive an explicit solution for the problem (1), where v = 0, 1,…, n.     

A condition of Schoenberg-Whitney type for multivariate spline interpolation

Lagrange interpolation by finite-dimensional spaces of multivariate spline functions defined on a polyhedral regionK in ^k is studied. A condition of Schoenberg-Whitney type is introduced. The main result of this paper shows that this condition characterizes all configurationsT inK such that in every neighborhood ofT inK there must exist a configuration which admits unique Lagrange interpolation.     

Chordal cubic spline interpolation is fourth-order accurate

^** Email: michaelf{at}ifi.uio.no It is well known that complete cubic spline interpolation offunctions with four continuous derivatives is fourth-order accurate.In this paper we show that this kind of interpolation, whenused to construct parametric spline curves through sequencesof points in any space dimension, is again fourth-order accurateif the parameter intervals are chosen by chord length. We alsoshow how such chordal spline interpolants can be used to approximatethe arc-length derivatives of a curve and its length.     

Optimal shift parameters for periodic spline interpolation

Using the exponential Euler spline, restricted on the unit circle, we sketch a unified approach to the periodic spline interpolation with shifted interpolation nodes. Mainly we are interested in the optimal choice of the shift parameter such that the corresponding interpolatory matrix possesses minimal condition or such that the related interpolation operator has minimal norm. We show that =0 is optimal in both cases. This improves known results of Merz, Reimer-Siepmann and Richards.     

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.     

Computational estimation of the constant

The paper describes a computational estimation of the constant characterizing the bounds of . It is known that as

with , while the truth of the Riemann hypothesis would also imply that . In the range , two sets of estimates of are computed, one for increasingly small minima and another for increasingly large maxima of . As increases, the estimates in the first set rapidly fall below and gradually reach values slightly below , while the estimates in the second set rapidly exceed and gradually reach values slightly above . The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.

     

On the quality of algorithms based on spline interpolation

The qualityq of a numerical algorithm using some specified information is the ratio of its error to the smallest possible error of an algorithm based on the same information. We use as information function values at equidistant points, periodicity and a bound for therth derivative. We show thatq is rather small, if the algorithm is based on spline interpolation.     

Lipscomb's space

It is known that Lipscomb's space can be imbedded in Hilbert's space . Let be the imbedded version of endowed with the -induced topology. We show how to construct as the attractor of an iterated function system containing an infinite number of affine transformations of . In this way we answer an open question of J.C. Perry.

     

Error estimates for discrete spline interpolation: Quintic and biquintic splines

In this paper we shall develop a class of discrete spline interpolates in one and two independent variables. Further, explicit error bounds in _?∞${?}_{\infty }$ norm are derived for the quintic and biquintic discrete spline interpolates. We also present some numerical examples to illustrate the results obtained.