非对角二元二次函数逼近的存在性和局部性质

§ 1　IntroductionThis paper is concerned with the properties of the simple off-diagonal bivariatequadratic Hermite-Padé approximation.Thisapproximation may be defined asfollows(see,for example,[1 ] ) .Let f(x,y) be a bivariate function,analytic in some neighbourhood of the origin(0 ,0 ) ,whose series expansion about the origin is known.Let a0 (x,y) ,a1 (x,y) ,a2 (x,y) bebivariate polynomials,a0 (x,y) = ki,j=0 a(0 )ij xiyj,a1 (x,y) = ni,j=0 a(1 )ij xiyj,a2 (x,y) = mi,j=0 a(2 )ij xiyj,such th…     

一类Ⅱ型剖分下的二元三次周期样条的超限插值和逼近(英)

本文讨论了Ⅱ一型三角剖分△^（2）_mn下的一类二元三次周期样条的超限插值和逼近,给出了它的表示以及存在唯一性,最后,估计了它的逼近阶．     

两个素数和一个素数平方和的丢番图逼近(英文)

We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + λ2 p2 + λ3 p2 3 + η| < (max pj )- 1/40 (log max pj ) 4 .     

二元二次对角逼近的逼近阶

给出了二元二次对角逼近的逼近阶     

二元Baskakov算子的逼近

在［１］中构造了一系列一元及多元线性算子，其中包括二元Ｂａｓｋａｋｏｖ算子，本文讨论该算子在Ｃ空间的逼近性质．     

关于随机padé逼近的存在性和收敛性

首次对随机Pade逼近进行了研究。考虑了随机形式幂级数 f(z,ω)=a_0ξ_0(ω)+a_iξ_i(ω)z+…(1)的Pade逼近。其中a_i(i=0,1,…)是全不为零的实数序列,ξ_i(ω)是独立的连续随机变量。 首先证明了(1)的任意Pade逼近的a.s.存在性。其次,考虑了一类形如 (2) 的随机准解析函数Pade逼近的a.s.依勒贝格测度收敛性。     

最佳L_∞逼近的存在性

令V_n=span{_1,_2,…,_n},设函数f∈L_p[E,μ],1≤p<∞,在点p 处定义一个最佳L_p 逼近算子τ∫(p)。记N_f(p)=∥f-τ∫(p)∥_p=inf/Q∈V_n∥f-Q∥_(po)本文证明了N_f(p)/[μ(E)]l/p 是p 的单调增加且有界的函数。如果f∈L_∞[E,μ],则存在τ∫(∞)∈V_n,使得∥f-τ∫(∞)∥_∞=inf/Q∈V_n∥f-Q∥∞,并且给出了最佳逼近值。     

The Existence and Local Behavior of Multivariate Algebraic Function Approximations

The existing scheme of rational polynomial approximants, defined by multivariate power series, is extended to define approximants with branch points. The existence theorem is obtained. The basic properties used to define the rational approximants can be preserved almost intactly. Especially, the local behavior of the     

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.     

一类二元有理插值函数的存在性及算法

文[3]构造了对于矩形网格上基于二元Newton插值公式的一类二元有理插值函数,并给出了其存在性的充分条件.本文进一步证明了这类二元有理插值函数存在性的必要条件,特别地,当m=n时,给出了具有三角形结构的系数矩阵的判别方法,该方法计算简便且具有承袭性,文章最后给出的实例说明了方法的有效性.     

Asymptotic Distributions of Zeros of Quadratic Hermite–Pade Polynomials Associated with the Exponential Function

The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e} polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$. The polynomials are defined by the relation $$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$ and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials $p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials $p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results obtained in [17].     

Bivariate segment approximation and splines

The problem to determine partitions of a given rectangle which are optimal for segment approximation (e.g., by bivariate piecewise polynomials) is investigated. We give criteria for optimal partitions and develop algorithms for computing optimal partitions of certain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain good partitions for interpolation by tensor product spline spaces. Our numerical examples show that the methods work efficiently.     

Convex Quadratic Approximation

For some applications it is desired to approximate a set of m data points in ⁿ with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n + 1)(n + 2)/2 parameters, s m, so as to minimize the approximation error in the L ¹ norm. The approximating function is q(p,x), where p ^s is the vector of parameters, and x n. The Hessian of q(p,x) with respect to x (for fixed p) is positive semi-definite, and its Hessian with respect to p (for fixed x) is shown to be positive semi-definite and of rank n. An algorithm is described for computing an optimal p* for any specified set of m data points, and computational results (for n = 4,6,10,15) are presented showing that the optimal q(p*,x) can be obtained efficiently. It is shown that the approximation will usually interpolate s of the m data points.     

On the Approximation of an Analytic Function Represented by Laplace-Stieltjes Transformation

In the present paper,we have considered the approximation of analytic functions represented by Laplace-Stieltjes transformations using sequence of definite integrals. We have characterized their order and type in terms of the rate of decrease of E_n(F,β) where E_n(F,β) is the error in approximating of the function F(s) by definite integral polynomials in the half plane Res≤βα.     

二元函数中值定理的注记

讨论了二元函数中值定理中间值的渐近性质,给出了一个相关反问题的解.     

Pole-Based Approximation of the Fermi-Dirac Function

Two approaches for the efficient rational approximation of the Fermi-Dirac function are discussed： one uses the contour integral representation and conformal real〉 ping, and the other is based on a version of the multipole representation of the Fermi-Dirac function that uses only simple poles. Both representations have logarithmic computational complexity. They are of great interest for electronic structure calculations.