Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Smooth Universal Taylor Series

Let be a simply connected domain in , such that is connected. If g is holomorphic in Ω and every derivative of g extends continuously on , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and we denote . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

On the Radial Behavior of Universal Taylor Series

In this work we deal with universal Taylor series in the open unit disk, in the sense of Nestoridis; see [12]. Such series are not (C,k) summable at every boundary point for every k; see [7], [11]. In the opposite direction, using approximation theorems of Arakeljan and Nersesjan we prove that universal Taylor series can be Abel summable at some points of the unit circle; these points can form any closed nowhere dense subset of the unit circle.     

Universal Taylor Series on General Doubly Connected Domains

In this paper, a proof is given that there exists a universalTaylor series – in the sense of Nestoridis – inthe complement of a disk with respect to every center. Thisanswers a question asked by G. Costakis. 2000 Mathematics SubjectClassification 30B30.     

Existence of Universal Taylor Series for Nonsimply Connected Domains

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂΩ that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.     

Boundary Behavior of Universal Taylor Series and Their Derivatives

For a given first category subset E of the unit circle and any given holomorphic function g on the open unit disk, we construct a universal Taylor series f on the open unit disk, such that, for every n = 0,1,2,..., f⁽ⁿ⁾ is close to g⁽ⁿ⁾ on a set of radii having endpoints in E. Therefore, there is a universal Taylor series f, such that f and all its derivatives have radial limits on all radii with endpoints in E. On the other hand, we prove that if f is a universal Taylor series on the open unit disk, then there exists a residual set G of the unit circle, such that for every strictly positive integer n, the derivative f⁽ⁿ⁾ is unbounded on all radii with endpoints in the set G.     

Banach Spaces of Universal Taylor Series in the Disc Algebra

We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no á priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form A + iB where A is self-adjoint, B is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrödinger and Dirac operators.     

A Universal Taylor Series in the Doubly Connected Domain C\{1}

We give a constructive proof of the existence of a universal Taylor series with center 0 in the doubly connected domain C\{1}. We also obtain a residuality result.     

随机Taylor级数

该文对一般的随机变量序列及相当弱的系数条件研究了随机级数定义的整函数的奈望里纳特征函数，并证明了它是几乎必然无有限例外值的．     

泰勒公式及泰勒级数之妙用

泰勒公式及泰勒级数是非常重要的数学工具,除了读者熟知的应用方面外,在其他问题的解决中也有妙用.举例介绍了应用泰勒公式及泰勒级数解决判断级数的敛散性、证明与积分有关的不等式等问题.     

Universal Fourier Series

Mathematical Notes -     

A Century of Interplay Between Taylor Series, Fourier Series and Brownian Motion

This article is an extended version of a lecture given in Oxfordon 12 May 1995 at the invitation of the London MathematicalSociety and the British Society for the History of Mathematics. Contents

1. A few figures
2. Taylor series before 1900. A strange statementof Borel
3. Fourier series before 1900. A strange field
4. Brownianmotion around 1900. A rising subject
5. Fourier and Taylor seriesafter 1900. A revival
6. Lacunarity and randomness
7. The appearanceof random series of functions
8. The Wiener theory of Brownianmotion
9. The merging of Brownian motion and random Fourier series
10. The non-differentiability and local behaviour of Brownianmotion
11. Three ways to figure out the Brownian motion
12. Theplane Brownian motion
13. Applications of Brownian motion to Taylorseries and analyticfunctions
14. Applications of Brownian motionto Fourier series and harmonicanalysis

1991 Mathematics Subject Classification 01A60, 30-03, 30B10,30B20, 42-03, 42A55, 42A61, 60-03, 60J65.     

关于Taylor级数与Fourier级数的几点注记

Taylor级数与Fourier级数是两类非常重要的函数项级数,二者在发展与应用背景、展开条件、收敛性和展开的唯一性等方面不尽相同,本文对此作了一些总结与探讨。     

关于Taylor级数的正规增长性

该文分别对有限级和无穷级Taylor级数的正规增长条件进行了系统的研究，取得了一系列新的结果．     

应用多元泰勒级数展开式计算电多极矩

对矩形排列且正负相间的4个点电荷,应用泰勒级数的三维展开式,可发现其电势由电多极矩所激发的各级电势叠加组成,从而得出其各级近似.     

Taylor Series in Hermitean Clifford Analysis

In this paper, we consider the Taylor decomposition for h-monogenic functions in Hermitean Clifford analysis. The latter is to be considered as a refinement of the classical orthogonal function theory, in which the structure group underlying the equations is reduced from mathfrakso(2m){mathfrak{so}(2m)}to the unitary Lie algebra u(m).