Adaptive bivariate Chebyshev approximation

We propose an adaptive algorithm which extends Chebyshev series approximation to bivariate functions, on domains which are smooth transformations of a square. The method is tested on functions with different degrees of regularity and on domains with various geometries. We show also an application to the fast evaluation of linear and nonlinear bivariate integral transforms. Work supported by the research project CPDA028291 “Efficient approximation methods for nonlocal discrete transforms” of the University of Padova, and by the GNCS-INdAM.     

矩形网格上的二元切触有理插值

二元切触有理插值是有理插值的一个重要内容,而降低其函数的次数和解决其函数的存在性是有理插值的一个重要问题.二元切触有理插值算法的可行性大都是有条件的,且计算复杂度较大,有理函数的次数较高.利用二元Hermite（埃米特）插值基函数的方法和二元多项式插值误差性质,构造出了一种二元切触有理插值算法并将其推广到向量值情形.较之其它算法,有理插值函数的次数和计算量较低.最后通过数值实例说明该算法的可行性是无条件的,且计算量低.     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

Optimizing an objective function under a bivariate probability model

The motivation of this paper is to obtain an analytical closed form of a quadratic objective function arising from a stochastic decision process with bivariate exponential probability distribution functions that may be dependent. This method is applicable when results need to be offered in an analytical closed form without double integrals. However, the study only applies to cases where the correlation coefficient between the two variables is positive or null. A stochastic, stationary objective function, involving a single decision variable in a quadratic form is studied. We use a primitive of a bivariate exponential distribution as first expressed by Downton [Downton, F., 1970. Bivariate exponential distributions in reliability theory. Journal of Royal Statistical Society B 32, 408–417] and revisited in Iliopoulos [Iliopoulos, George., 2003. Estimation of parametric functions in Downton’s bivariate exponential distribution. Journal of statistical planning and inference 117, 169–184]. With this primitive, optimization of objective functions in Operations Research, supply chain management or any other setting involving two random variables, or calculations which involve evaluating conditional expectations of two joint random variables are direct. We believe the results can be extended to other cases where exponential bivariates are encountered in economic objective function evaluations. Computation algorithms are offered which substantially reduce computation time when solving numerical examples.     

二元二次函数逼近的存在性和局部性

This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Hermite-Pade form always defines a bivariate quadratic function and that this function is analytic in a neighborhood of the origin.     

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.     

Monotonicity preservation on triangles

It is well known that Bernstein polynomials on triangles preserve monotonicity. In this paper we define and study three kinds of monotonicity preservation of systems of bivariate functions on a triangle. We characterize and compare several of these systems and derive some geometric applications.

     

二元向量值有理插值存在性的一种判别方法

文章给出了对于矩形网格上基于二元Newton插值公式的二元向量值有理插值存在性的充要条件.在存在的情况下,建立了具有显式表达式的不同于向量连分式的二元向量值有理插值函数,并且这种方法具有承袭性.最后给出的实例说明了这种算法的有效性.     

Classification of Bivariate Configurations with Simple Lagrange Interpolation Formulae

In 1977 Chung and Yao introduced a geometric characterization in multivariate interpolation in order to identify distributions of points such that the Lagrange functions are products of real polynomials of first degree. We discuss and describe completely all these configurations up to degree 4 in the bivariate case. The number of lines containing more nodes than the degree is used for classifying these configurations.     

一类二元函数连续与可微条件的归纳与推广

多元函数的可导性、连续性、可微性是多元函数微分学的重要内容.本文基于多元微分学教学实践,归纳了一类二元函数的可偏导、连续、可微的充分条件,并对此类二元函数做一般性推广,得到了推广函数连续与可微的充分条件.     

关于Hsler-Reiss分布的一点注记

本文研究了服从二元二项分布的随机向量序列的极值分布问题.利用构造性的方法,证明了服从二元二项分布的随机向量序列之极值依分布收敛到Hsler-Reiss 分布, 将已有结论推广到离散情形.     

二元函数极值问题的新评注

对二元函数的极值判定条件进行了新的补充分析,给出了临界情形下的又一充分条件,并做了简明的证明.     

一种求二元有理插值函数的方法

给出一种方法可直接计算基于矩形节点的二元有理插值函数的分母在节点处的值 ,进而判断相应的二元有理插值函数是否存在 .此方法运用灵活 ,适用范围广 ,在相应的有理插值函数存在时 ,能给出它的具体表达式 .此外 ,我们还针对文中两个主要逆矩阵 ,给出了相应的递推公式 ,避免了求逆计算 .     

Morgenstern族次序统计量的协变量的分布

本文研究了Morgenstern族次序统计量的协变量的分布及其在筛选测验中的应用.利用排序集抽样的方法,获得了分组测验情形下每组协变量的极值的分布.最后,将上述结果应用至总体为Gumbel二维指数分布和二维均匀分布的情形,给出了不同的样本量下选择问题的筛选效率.     

A bivariate stable characterization and domains of attraction

Bivariate stable distributions are defined as those having a domain of attraction, where vectors are used for normalization. These distributions are identified and their domains of attraction are given in a number of equivalent forms. In one case, marginal convergence implies joint convergence. A bivariate optional stopping property is given. Applications to bivariate random walk are suggested.     

GLOBAL SMOOTHNESS PRESERVATION BY BIVARIATE INTERPOLATION OPERATORS

1　IntroductionIn the case when Pn(f,x) represents the univariate interpolation polynomial of Her-mite-Fejér based on Chebyshev nodesof the firstkind or the univariate interpolation polyno-mials of Lagrange based on Chebyshev nodes of the second kind and± 1 ,or the univariaterational Shepard operators,the following result of partial preservation of global smoothnessis proved in[4] :If f∈Lip M(α;[-1 ,1 ] ) ,0 <α≤ 1 ,then there existsβ=β(α) <α and M′>such thatω(Pn(f ) ;h)≤ M′h…     

NEAR MINIMAX POLYNOMIAL APPROXIMATION

In this paper a representation for linear polynomial projection is given. Then it isshown that the near minimax approximation is equivalant to a problem of the best approxima-tion of hivariste functions. And some properties for near minimax are obtained.     

On the convergence of bivariate two-point Padé-type approximants

Some choices of denominators are given which ensure the geometrical convergence of certain convergence of bivariate two-point Padé-type approximants to functions being holomorphic on certain domains     

On a completeness property of series expansions of bivariate densities

Random processes passed through zero memory nonlinearities are considered in terms of the preservation of mean square continuity. Then, series expansions of bivariate densities in orthonormal functions are treated, under the assumption that the bivariate density is associated with a first order stationary random process. It is shown that, if the random process is mean square continuous, then the orthonormal functions in the series expansion must be complete.