On meromorphic approximation

Let G be a bounded N-connected domain, the boundary of which consists of closed analytic Jordan curves. We assume that 0G. For any nonnegative integers n and m, denote by _n,m the class of all meromorphic functions on G that can be represented in the form h=p/qz ^m , where p belongs to the Smirnov class E (G), q is a polynomial of degree at most n, q0. Theorems giving necessary and sufficient conditions for a function belonging to the class _n,m to be an element of best approximation to a continuous function f on in the space L () by functions in the class _n,m are proved. Some questions concerning orthogonal polynomials and the theory of Hankel operators are also considered.     

亚纯函数的唯一性定理

应用奈望里纳理论研究亚纯函数的唯一性问题,得到若干结果,推广了某些已知定理．     

Sufficient condition for functions to be in a class of meromorphic multivalent Sakaguchi type spiral-like functions

In this article, we define a subclass of meromorphic multivalent Sakaguchi type functions and obtain certain sufficient conditions for functions to be in this class. The main result presented here includes a number of consequences as its special cases.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

Uniform approximation by meromorphic functions on Riemann surfaces

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE ⁰ can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin. Partially supported by a grant from NSERC of Canada.     

Completely irreducible sequences of meromorphic functions

Given a sequence {fn } of meromorphic functions on a plane domain D, there exists a (possibly empty) open set U■D and a subsequence {f n k } which converges uniformly (with respect to the spherical metric on ) on compact subsets of U, while no subsequence of {f n k } converges uniformly on compact subsets of any larger open subset of D.     

CM分担三个公共小函数对的亚纯函数的惟一性

研究了具有三个CM公共小函数对，一个IM公共小函数对的亚纯函数的惟一性，改进了李平与杨重骏，Brosch等人的有关结果，用例子说明本文的结果是最佳的.     

具有两个权分担的公共值集的亚纯函数

讨论了亚纯函数分担两个集合的唯一性问题,证明了一个定理.所得结果改进了先前的一些定理,例子表明结果是精确的.     

On the family of meromorphic functions whose derivatives omit a holomorphic function

In this paper,we continue to study the normality of a family of meromorphic functions without simple zeros and simple poles such that their derivatives omit a given holomorphic function.Such a family in general is not normal at the zeros of the omitted function.Our main result is the characterization of the non-normal sequences,and hence some known results are its corollaries.     

Korovkin-type approximation in spaces of vector-valued and set-valued functions

We consider some Korovkin-type approximation results for sequences of linear continuous operators in spaces of vector-valued and set-valued continuous functions without assuming the existence of the limit operator. Even in spaces of real continuous functions, where similar results have already been established, we replace the positivity assumption with a weaker condition. We also give some quantitative estimate of the convergence and some applications where previous results cannot be applied.     

The size of the Julia set of meromorphic functions

We give a lower bound of the hyperbolic and the Hausdorff dimension of the Julia set of meromorphic functions of finite order under very general conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zeros and fixed points of difference operators of meromorphic functions

Let f be a transcendental meromorphic function and Δf(z) = f(z + 1) − f(z). A number of results are proved concerning the existences of zeros and fixed points of Δ f(z) and Δ f(z)/f(z) when f(z) is of order σ(f)=1. Examples show that some of the results are sharp.     

Gleason's problem and tangential homogeneous interpolation for hyperholomorphic quaternionic functions

We study a version of Gleason's problem in the setting of functions of class C¹ in the unit ball of ?². We use the setting of hyperholomorphic functions to define and solve the problem. Finally, we briefly discuss a tangential interpolation problem for hyperholomorphic functions.     

On multiply-connected Fatou components in iteration of meromorphic functions

Let be a transcendental meromorphic function with at most finitely many poles. We mainly investigated the existence of the Baker wandering domains of f(z) and proved, among others, that if f(z) has a Baker wandering domain U, then for all sufficiently large n, fn(U) contains a round annulus whose module tends to infinity as n→∞ and so for some 0<d<1,

     

An improved estimate of PSWF approximation and approximation by Mathieu functions

In this paper, an error estimate of spectral approximations by prolate spheroidal wave functions (PSWFs) with explicit dependence on the bandwidth parameter and optimal order of convergence is derived, which improves the existing result in [Chen et al., Spectral methods based on prolate spheroidal wave functions for hyperbolic PDEs, SIAM J. Numer. Anal. 43 (5) (2005) 1912-1933]. The underlying argument is applied to analyze spectral approximations of periodic functions by Mathieu functions, which leads to new estimates featured with explicit dependence on the intrinsic parameter.     

Copositive approximation by rational functions with prescribed numerator degree

The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Jackson type estimate ｜｜f-r｜｜p≤Cδl^2μωφ（f,1/n）p holds, where μ is a natural number ≥3/2＋1/p, and Cδ is a positive constant depending only on δ.