Spherical spline interpolation—basic theory and computational aspects

The purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in Euclidean spaces. Spline functions are defined on the sphere. The solution process is made simple and efficient for numerical computation. In addition, the convergence of the solution obtained by spherical spline interpolation is developed using estimates for Legendre polynomials.     

Bivariate Polynomial Natural Spline Interpolation to Scattered Data

By means of the theory of spline interpolation in Hilbert spaces, the bivariate polynomial natural spline interpolation to scattered data is constructed. The method can easily be carried out on a computer, and parallelly generalized to high dimensional cases as well. The results can be used for numerical integration in higher dimensions and numerical solution of partial differential equations, and so on.     

Weak descartes systems in generalized spline spaces

A class of generalized spline spaces is introduced for which a basis of functions with local support is constructed by using a recursion relation. It is shown that this basis forms a weak Descartes system. Moreover, an interpolation property is given.     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。     

散乱数据的多项式自然样条光顺与广义插值

由于理论与实践的重要性,在多元插值方面有相当多的工作,如[1]-[11]。目前以箱样条(box splines),光滑余因子与B网方法以及薄板样条与径函数(radial basis function)方法比较活跃。前者具有良好的性质和丰富的结构,很快成为一个活跃的研究方向,最近更在小波(wavelet)变换理论研究上发挥了作用。但是,它一般只处理规则分划的问题,不能做多元散乱数据的插值。     

样条空间S~μ_k（△）的维数

本文考虑了欧式空间Ｒ￣ｎ中任意单纯形剖分上的样条函数空间．证明了当ｋ≥（３μ＋１）２￣（ｎ－２）＋１时，计算任意单纯形剖分Δ上的ｋ次μ阶光滑样条空间的维数，可归结为计算每个σ－关联域（ｉ－单纯形σ∈Δ）Ｒ（σ）上的２￣（ｎ－ｉ－１）μ次μ阶光滑（ｉ≤ｎ－１）样条空间的维数。这里σ－关联域Ｒ（σ）是指Δ中所有包含σ的单纯形所成的单纯形剖分．     

Which spaces for design?

We determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms. It is the class of all spaces containing constants of which the spaces derived under differentiation are Quasi Extended Chebyshev spaces, i.e., they permit Hermite interpolation, Taylor interpolation excepted. It is also the class of all spaces which possess Bernstein bases, or of all spaces for which any associated spline space does possess a B-spline basis. Note that blossoms guarantee that such bases are normalised totally positive bases. They even are the optimal ones.     

A class of multidimensional periodic splines

Periodic spline functions are introduced by use of reproducing kernel structure in Hilbert spaces. Minimum properties are described in interpolation and best approximation problems. A numerical method for determining interpolating splines and best approximations is proposed.Dedicated to Prof. Dr. F. Reutter on the occasion of his 70th birthday     

W_2~m空间中样条插值算子与线性泛函的最佳逼近

In this paper,the convergency of spline interpolation operators is obtained,these spline operators are determined by linear differential operators and constraint functionals.The errors of the interpolating spline with EHB fanctionals are estimated.The best approximation of linear functionals on W2^n spaces are investigated,which let to a useful computational method for the approximation solution of higher order linear differential equations with multipoint boundary value conditions.     

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.     

Free interpolation of germs of analytic functions in Hardy spaces

One proves theorems on the interpolation of germs of analytic functions, defined in the neighborhoods of the interpolation nodes, in the Hardy spaces H^P(0 < p +), generalizing the corresponding results of N. K. Nikol'skii and V. I. Vasyunin for the classes H and H². One obtains estimates of the norms of the interpolating functions in terms of the parameter of the set on which the interpolation is performed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 36–45, 1982.     

Lagrange and hermite interpolation by bivariate splines

We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.     

Macro-elements and stable local bases for splines on Powell-Sabin triangulations

Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundary-value problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.

     

A remark on interpolation by bivariate splines

We study the determining set for bivariate spline spacesS _k ^o on type-1 triangulation of square using B-net techniques. We further construct the interpolation schemes for these spline spaces that are unisolvent for any function f of C^σ.     

W-加密三角剖分下二元五次超样条函数空间的局部Lagrange插值

本文选取二元五次C2超样条函数空间作为插值空间,考虑局部Lagrange插值.首先对三角剖分△进行着色,通过Wang-加密三角剖分对原剖分△细分大约一半的三角形.然后通过在内边增加一些另外的光滑条件,使得样条函数在某些边上达到更高阶的光滑.最后在△的加密三角剖分内选择Lagrange插值点.结果表明相应的插值基函数具有...     

形状可调的C~2连续三次三角Hermite插值样条

基于函数空间{1,sint,cost,sin~2t,sin~3t,cos~3t}构造了一种形状可调的三次三角Hermite插值样条.该样条不仅具有带参数的Hermite型插值样条的主要特性,而且在插值节点为等距时可自动满足C2连续,其形状还可通过所带的参数进行调节.在适当条件下,该样条对应的Ferguson曲线可精确表示工程中一些常见的曲线.     

Bivariate spline interpolation at grid points

Summary. We describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are selected in such a way that the grid points of the partition are contained in these sets, and no large linear systems have to be solved. Our method is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied. In order to include the grid points in the interpolation sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This approach is completely different from the known interpolation methods for bivariate splines of degree at most three. Our method is illustrated by some numerical examples. Received October 5, 1992 / Revised version received May 13, 1994     

Macro-elements and stable local bases for splines on Clough-Tocher triangulations

Summary. Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving boundary-value problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Clough-Tocher refinements of arbitrary triangulations. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. Received November 18, 1999 / Published online October 16, 2000     

Nöther-type theorem of piecewise algebraic curves on quasi-cross-cut partition

Nöther’s theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nöther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.