An asymptotically hierarchy-consistent, iterative sequence transformation for convergence acceleration of Fourier series

We derive the I transformation, an iterative sequence transformation that is useful for the convergence acceleration of certain Fourier series. The derivation is based on the concept of hierarchical consistency in the asymptotic regime. We show that this sequence transformation is a special case of the J transformation. Thus, many properties of the I transformation can be deduced from the known properties of the J transformation (like the kernel, determinantal representations, and theorems on convergence behavior and stability). Besides explicit formulas for the kernel, some basic convergence theorems for the I transformation are given here. Further, numerical results are presented that show that suitable variants of the I transformation are powerful nonlinear convergence accelerators for Fourier series with coefficients of monotonic behavior. This revised version was published online in August 2006 with corrections to the Cover Date.     

与中心数相关的一类级数求和公式

This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.     

广义四元数矩阵方程AX-DXB=R

设HF为域F上的广义四元数除环，ChF≠2。本文利用拟线性变换T（X）＝AX－DXB讨论HF上矩阵方程AX－DXB＝R的求解问题，获得了上方程存在（唯一）解的几个充分必要条件，并给出了解的显式公式。     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN Cn

Some previous results on convergence of Taylor series in Cn [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in Cn are constructed and the Taylor series expansion is deduced.     

Summation of Some Trigonometric and Schlömilch Series

In this paper we consider trigonometric series in terms of the Riemann zeta function and related functions of reciprocal powers. The obtained closed form formulas we apply to the evaluation of the Riemann zeta function and related functions of reciprocal powers. One can establish recursive relations for them and relations between any two of those functions. These closed formulas enable us also to find sums of some Schlömilch series. We give an example which shows how the convergence of a trigonometric series can be accelerated by applying Krylov's method and our formula (7).     

Uniform Convergence of Fourier–Jacobi Series

Necessary and sufficient conditions which imply the uniform convergence of the Fourier–Jacobi series of a continuous function are obtained under an assumption that the Fourier–Jacobi series is convergent at the end points of the segment of orthogonality [−1,1]. The conditions are in terms of the modulus of continuity, Λ-variation, and the modulus of variation of a function.     

一个含四个独立基的变换公式及其应用

本文利用反演的方法得到了一个四个独立基的变换公式并由此得到了几个新的基本超几何级数求和公式和超几何级数求和公式.     

Transformation of the triple series of Gelfand, Graev, and Retakh into a series of the same type and related problems

A transformation of the triple series T related to the GrassmanianG _2,4 into a series of the same structure type is obtained. This transformation generalizes the reduction formula of Gelfand, Graev, and Retakh taking the series T to the Gauss function under two additional conditions and two more general reduction formulas taking the series T to the Appell function F ₁ and to the Horn function G ₂ under one of the additional conditions. The approach used to analyze the series T is based on the representation of the initial series T in terms of series with convenient computational properties.     

A converse of the mean value theorem for integrals

We derive two simple and memorizable formulas for the radius of convergence of a power series which seem to be appropriate for teaching in an introductory calculus course.     

Error estimate of the series solution to a class of nonlinear fractional differential equations

In this paper, two different formulas for the accelerated Adomian polynomials are considered. The discussion demonstrates that both formulas are identically the same and provide the fastest rate of convergence for the nonlinear term. One of the two formulas, suggested by the author, is used directly to estimate the maximum absolute truncated error of the Adomian series solution.     

The practical use of the Euler transformation

The generalised Euler transformation is a powerful transformation of infinite series which can be used, in theory, for the acceleration of convergence and for analytic continuation. When the transformation is applied to a series with rounded coefficients, its behaviour can differ substantially from that predicted theoretically. In general, analytic continuation is impossible in this case. It is still possible, however, to use the transformation for acceleration of convergence, but some changes are necessary in the method of choosing the optimum parameter value.     

A Levin-type algorithm for accelerating the convergence of Fourier series

A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincaré-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm.     

关于一个普遍本源公式及其应用

Here presented is a further investigation on a general source formula(GSF) that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author's recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31(3):260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308:3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the(Σ?D) formula class are given in the closing section.     

A symbolic operator approach to several summation formulas for power series II

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.     

A certain class of q-series transformations

A large number of summation and transformation formulas for a certain class of double hypergeometric series are observed here to follow fairly readily from a single known result which, in turn, is a very special case of one of six general expansion formulas given in the literature. Generalizations and unifications of these expansion formulas involving series with essentially arbitrary terms are presented. It is also shown how the various series transformations considered in this paper admit themselves of $\text{q}$-extensions which are capable of unifying numerous scattered results in the theory of basic double hypergeometric functions.     

The convergence of fourier series in eigenfunctions of the Schrödinger operator with Kato potential

We obtain sharp conditions for the absolute uniform convergence of Fourier series in the eigenfunctions of the Schrödinger operator with Kato potential in a bounded domain for functions lying in the domains of generalized fractional powers of the original Schrödinger operator or in generalized Besov classes with a sharp exponent.     

富里叶级数的收敛速度

对正弦和余弦富立叶级数,通过合并相邻同号项,使其重排成交错级数.讨论了重排形成的交错级数的敛散性.指出根据自变量x的不同取值,该交错级数可能是单调递减或周期递减的级数.按照莱布尼茨判定法提出了不同精度要求的级数项数的计算公式.选取一到三阶收敛的富立叶级数计算了不同比值精度及差值精度要求的级数项数.计算表明,在x的取值为2π的等分点时,富立叶级数的部分和随项数的增加单调地逼近其收敛值.在x的取值为其它点时,富立叶级数的部分和随项数的增加围绕收敛值上下变动,周期地逼近其收敛值.低收敛阶富立叶级数的收敛速度较慢.要达到0.01%的精度,一收敛阶富立叶级数需要数万项,二收敛阶富立叶级数也需要数百项.在不同计算点处,要达到相同的计算精度,需要的级数项数差别较大.     

Series transformation formulas of Euler type,Hadamard product of series,and harmonic number identities

The Hadamard multiplication theorem for series is used to establish several Euler-type series transformation formulas. As applications we obtain a number of binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers.     

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.