Interpolation Formula for Functions of Several Variables

Mathematical Notes -     

On Extremal Properties of Interpolation Polynomials for Functions of Several Variables

By using earlier results, we find solutions of two extremal problems on sets of interpolation polynomials for functions of several variables.     

多复变量亚纯函数的第二基本定理和唯一性定理

本文给出了多复变量亚纯函数的第二基本定理的一个改进形式,并进一步得到了一个关于重值的唯一性定理(详见文[1-16])。     

关于多元幂级数

引入了多元函数项级数的概念,给出了其收敛域及和函数的定义;通过详实的例子讨论了多元幂级数的收敛域、和函数及多元函数展开为多元幂级数的计算方法.     

多元函数的可微性

给出了多元函数可微性的等价条件 ;利用这个条件可以简便有效地判别多元函数的可微性     

Rational Interpolation for Stieltjes Functions

The continuity conclusions about rational Hermite interpolating functions are given under some conditions. On that basis, we establish the convergence results for the paradiagonal sequences of the rational interpolants for Stieltjes functions and Hamburger functions.     

On Some Properties of Polynomial Bases of Subspaces over the Field of Rational Functions in Several Variables

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a free polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.     

On Chebyshev–Markov Rational Functions over Several Intervals

Chebyshev–Markov rational functions are the solutions of the following extremal problem

Full-size image

withKbeing a compact subset of andω_n(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b₁, b₂]…[b_2p−1, b_2p], −∞<b₁b₂<…<b_2p−1b_2p<+∞ in terms of Schottky–Burnside automorphic functions.     

关于多元调和函数论中的唯一性定理

我们感兴趣的是将自变量为２元时成立的某些较深刻的关于共轭调和函数系中的唯一性定理推广到ｎ元的情形。当ｎ≤１２时,它们不仅在上半空间Ｒ＾ｎ＋１＋上成立,而且在更为一般的区域Ω真包含于Ｒ＾ｎ＋１上也成立。从而证明了Ｔ．Ｈ．Ｗｏｌｆｆ三大反例中的第一大反例是错误的。     

On the Nonrepresentability of Functions of Several Variables by Quadratures

The paper completes the construction of a multidimensional topological version of differential Galois theory. We construct a rich class of germs of functions of several variables which is closed under superpositions and other natural operations. The main theorem describes the behavior of the monodromy groups of such germs under the natural operations. As a result, we obtain topological obstructions to the representability of functions by quadratures, which give the strongest known statements about unsolvability of equations in closed form.     

多个亚纯函数的分担值与唯一性

本文研究了多个亚纯函数的分担值与唯一性问题,得到了一般情况下相应的唯一性定理。     

On Meromorphic Starlike Multivalent Functions

In this paper we introduce and study some new subclasses of meromorphic starlike multivalentfunctions.Inclusion relations are established,Integral transforms of functions in these classes are alsoconsidered.In particular,our results include or improve several results due to Mogra et al.[2],Mogra[3],Goel and Sohe[4]and Bajpai[5].     

Degeneracy and Finiteness Theorems for Meromorphic Mappings in Several Complex Variables

The author proves that there are at most two meromorphic mappings of C~m into P~n(C)(n ≥ 2) sharing 2 n+ 2 hyperplanes in general position regardless of multiplicity,where all zeros with multiplicities more than certain values do not need to be counted. He also shows that if three meromorphic mappings f~1, f~2, f~3 of Cminto P~n(C)(n ≥ 5) share2 n+1 hyperplanes in general position with truncated multiplicity, then the map f~1×f~2×f~3 is linearly degenerate.     

伪多元函数的Thiele型有理插值

根据伪多元函数的特性,主要讨论了伪多元函数的Thiele型有理插值,并给出了Thiele型有理插值的误差.     

On Approximation Orders of Functions of Several Variables in the Lorentz Space

We consider the anisotropic Lorentz space of periodic functions. Sufficient conditions are proved for a function to belong to the anisotropic Lorentz space. Estimates for the order of approximation by trigonometric polynomials of the Nikol’skii–Besov class in the anisotropic Lorentz space are established.     

On Rational Interpolation to |x|

We consider Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes, and we show that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. Date received: August 18, 1995. Date revised: January 10, 1996.     

给定极点的有理函数插值序列的收敛性

设Γ∈Ｃ（１，α），α＞０．Ｇ是复平面上以Γ为边界的有界单连通区域．本文考虑了极点位于G外部，以广义Ｆａｂｅｒ-Ｄzｒｂａsｊａｎ有理函数的零点为插值结点的Ｌａｇｒａｎｇｅ插值有理函数序列对Ａ（G）和Ｅ^ｑ（Ｇ）（１＜ｑ＜＋∞）中函数的一致逼近和平均逼近阶的估计．     

Bloch Functions in Several Complex Variables, I

Bloch functions on the unit disk are those analytic functionsf for which the quantity |f'(z)|(1 – |z|²) is bounded(for z in the disk). In this paper, Bloch functions on boundedhomogeneous domains in complex m-space are studied. The mainresult shows that many of the equivalent definitions of Blochfunctions on the unit disk are also equivalent in the generalsetting. Current Address: 39 Trinity College, Dublin 2, Ireland.