Explicit Expression of Hermite Interpolation Basis Functions under the Ball Basis

In paper [1],it was shown that an explicit expression of the cardinal basis functions for two-point Hermite interpolation. This paper will show the explicit expression of Hermite interpolation under the Ball basis.     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

On multivariate Hermite interpolation

We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.     

About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval

We obtain the Laurent polynomial of Hermite interpolation on the unit circle for nodal systems more general than those formed by the n-roots of complex numbers with modulus one. Under suitable assumptions for the nodal system, that is, when it is constituted by the zeros of para-orthogonal polynomials with respect to appropriate measures or when it satisfies certain properties, we prove the convergence of the polynomial of Hermite-Fejér interpolation for continuous functions. Moreover, we also study the general Hermite interpolation problem on the unit circle and we obtain a sufficient condition on the interpolation conditions for the derivatives, in order to have uniform convergence for continuous functions.Finally, we obtain some improvements on the Hermite interpolation problems on the interval and for the Hermite trigonometric interpolation.     

关于球面上的一种Hermite插值

本文旨在提出单位球面上的一种Hermite插值格式.为此,本文首先研究了沿球面同轴圆周组上的Hermite插值问题,给出了三种适定的插值泛函组.然后研究了球面上的Hermite插值问题,给出了球面上Hermite插值的一种叠加插值法,即添加圆周组法.进一步将二者结合,导出了一类球面上Hermite插值的适定插值泛函组.为了说明这类适定插值泛函组的构造方法,在本文最后还给出了构造球面上低次Hermite适定插值泛函组的一些具体例子和数值算例.     

Construction of Interpolating Scaling Vectors with Hermite Interpolation Property

In this paper, we construct a new family of Hermite-type interpolating scaling vectors with compact support, of which the Hermite interpolation property generalizes the existing results of interpolating scaling vectors and Hermite interpolants. In terms of the Hermite interpolatory mask, we characterize the Hermite interpolation property, approximation property and symmetry property in detail. To illustrate these results, several examples with compact support and high smoothness are exhibited at the end of this paper.     

Harmonic interpolation of Hermite type based on Radon projections with constant distances

We study harmonic interpolation of Hermite type of harmonic functions based on Radon projections with constant distances of chords. We show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords coalesce to some points on the unit circle, we prove that the interpolation polynomials tend to a Hermite interpolation polynomial at the coalescing points.     

Newton basis for multivariate Birkhoff interpolation

Multivariate Birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far. In this paper, we introduce a special new type of multivariate Birkhoff interpolation and present a Newton paradigm for it. Using the algorithms proposed in this paper, we can construct a Hermite system for any interpolation problem of this type and then obtain a Newton basis for the problem w.r.t. the Hermite system.     

Algorithms for solving Hermite interpolation problems using the Fast Fourier Transform

We present a method for computing the Hermite interpolation polynomial based on equally spaced nodes on the unit circle with an arbitrary number of derivatives in the case of algebraic and Laurent polynomials. It is an adaptation of the method of the Fast Fourier Transform (FFT) for this type of problems with the following characteristics: easy computation, small number of operations and easy implementation.In the second part of the paper we adapt the algorithm for computing the Hermite interpolation polynomial based on the nodes of the Tchebycheff polynomials and we also study Hermite trigonometric interpolation problems.     

一类多元全次数 Hermite 插值问题研究

In this paper we study a class of multivariate Hermite interpolation problem on 2~d nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combinatorial identities are firstly given and then the regularity of the proposed interpolation problem is proved.     

带二阶导数的插值定理

用B_(δ,p)(1P∞)表示P次可积Fourier变換具有紧支集[-δ,δ]的带有限函数。证明了对于B_(3δ,p)中的函数,可以在L_p(R)尺度下,由序列{f(κπ/δ)},{f'(κπ/δ)}以及{f'(κπ/δ)}的Hermite型插值进行重构.     

General interpolation on the lattice h{\Bbb Z}^s: Compactly supported fundamental solutions

Summary. In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions of compactly supported solutions for cardinal interpolation to gain compactly supported fundamental solutions for the general interpolation problem. The general interpolation problem admits the interpolation of the functional and derivative values under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructions of compactly supported fundamental solutions for cardinal interpolation are discussed together with algorithms for their construction that make use of MAPLE. Another construction based on finite decomposition and reconstruction for spline spaces is also provided. Ideas used in the latter construction are lifted to provide a general construction of compactly supported fundamental solutions for cardinal interpolation in the multivariate case. Examples are provided, several in the context of some general interpolation problem to illustrate how easy is the transition from cardinal interpolation to general interpolation. Received May 11, 1993 / Revised version received August 16, 1994     

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines S_q^r(Δ¹) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.     

C~1连续的双3次双4次Hermite元各向异性分析

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A generalized Taylor factorization for Hermite subdivision schemes

In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so-called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of a generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity.     

样条型矩阵有理插值

The matrix valued rational interpolation is very useful in the partial realization problem and model reduction for all the linear system theory. Lagrange basic functions have been used in matrix valued rational interpolation. In this paper, according to the property of cardinal spline interpolation, we constructed a kind of spline type matrix valued rational interpolation, which based on cardinal spline. This spline type interpolation can avoid instability of high order polynomial interpolation and we obtained a useful formula.     

A generalized Taylor factorization for Hermite subdivision schemes

In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so-called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of a generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity.     

一类收敛的线性Hermite重心权有理插值

众所周知, Hermite有理插值比Hermite多项式插值具有更好的逼近性, 特别是对于插值点序列较大时, 但很难解决收敛性问题和控制实极点的出现. 本文建立了一类线性Hermite重心有理插值函数$r(x)$,并证明其具有以下优良性质: 第一, 在实数范围内无极点; 第二, 当$k=0,1,2$时,无论插值节点如何分布, 函数$r^{(k)}(x)$具有$O(h^{3d+3-k})$的收敛速度; 第三, 插值函数$r(x)$仅仅线性依赖于插值数据.     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.