On a general class of trigonometric functions and Fourier series†

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented.     

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.     

三角级数求和的两种方法

借助实例介绍如何利用傅立叶级数和复变函数的幂级数这两种工具解决有关三角级数的求和问题.     

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.     

广义调和级数的推广

将广义调和级数sum from n=1 to ∞ 1/n~x推广为一类指数项级数sum from n=1 to ∞ a_nd_n~x,并证明了这类指数项级数有结构简单的收敛域,其和函数的性质与幂级数的相似.     

无限级Dirichlet级数及随机Dirichlet级数

主要研究全平面上无限级Ｄｉｒｉｃｈｌｅｔ级数及随机Ｄｉｒｉｃｈｌｅｔ级数的增长性．对于 Ｄｉｒｉｃｈｌｅｔ级数,研究了它的增长性和正则增长性,得到了它的系数和指数与增长级的两 个充要条件．对于随机Ｄｉｒｉｃｈｌｅｔ级数,证明了它的增长性几乎必然与其在每条水平直线 上的增长性相同．     

Extrapolation Algorithms for Filtering Series of Functions,and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the -algorithm) or by approximating the series by a rational function (in particular, a Padé approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.     

Comparison of two orthogonal series methods of estimating a density and its derivatives on an interval

We compare the merits of two orthogonal series methods of estimating a density and its derivatives on a compact interval—those based on Legendre polynomials, and on trigonometric functions. By examining the rates of convergence of their mean square errors we show that the Legendre polynomial estimators are superior in many respects. However, Legendre polynomial series can be more difficult to construct than trigonometric series, and to overcome this difficulty we show how to modify trigonometric series estimators to make them more competitive.     

Series solutions of a semilinear wave equation

We are concerned with the reconstruction of series solutions of a semilinear wave equation with a quadratic nonlinearity. The solution which may blow up in finite time is sought as a sum of exponential functions and is shown to be a classical one. The constructed solutions can be used to benchmark numerical methods used to approximate solutions of nonlinear equations.     

Asymptotic upper bounds for the coefficients in the Chebyshev series expansion for a general order integral of a function

The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the integral of a function. The tightness of this upper bound is then analyzed for the case , the first integral of a function. It is shown that for geometrically converging Chebyshev series the theorem gives the tightest upper bound possible as . For functions that are singular at the endpoints of the Chebyshev interval, , the theorem is weakened. Two examples are given. In the first example, we apply the method of steepest descent to directly determine (laboriously!) the asymptotic Chebyshev coefficients for a function whose asymptotics have not been given previously in the literature: a Gaussian with a maximum at an endpoint of the expansion interval. We then easily obtain the asymptotic behavior of its first integral, the error function, through the application of the theorem. The second example shows the theorem is weakened for functions that are regular except at . We conjecture that it is only for this class of functions that the theorem gives a poor upper bound.

     

On Fourier series for higher order (partial) derivatives of functions

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimates of efficiency for two methods of stable numerical summation of smooth functions

We consider a classical ill-posed problem of reconstruction of continuous functions from their noisy Fourier coefficients. We study the case of functions of two variables that has been much less investigated. The smoothness of reconstructed functions is measured in terms of the Sobolev classes as well as the classes of functions with dominated mixed derivatives. We investigate two summation methods, that are based on ideas of the rectangle and the hyperbolic cross, respectively. For both of these methods we establish the estimates of the accuracy on the classes that are considered as well the estimates of computational costs. Moreover, we made the comparison of their efficiency based on obtained estimates. A somehow surprising outcome of our study is that for both types of the considered smoothness classes one should employ hyperbolic cross approximation that is not typical for the functions under consideration.     

Determination of jumps for functions via derivative Gabor series

Recently, Shi Xianliang and Hu Lan published the method of concentration factors for determination of jumps of functions via MCM conjugate wavelets. Usually, it is difficult to calculate the Hilbert transform of general window functions. The aim of this paper is to discuss determination of jumps for functions based on derivative Gabor series. The results will simplify the calculation of jump values.     

有限级Dirchlet级数及随机Dirchlet级数

本文研究了全平面上有限级Dirichlet级数的增长性和正规增长性,得到了两个充要条 件;证明了有限级随机Dirichlet级数的增长性几乎必然与其在每条水平直线上的增长性相同．     

DEFICIENT FUNCTIONS OF RANDOM DIRICHLET SERIES

In this article, the uniqueness theorem of Dirichlet series is proved. Then the random Dirichlet series in the right half plane is studied, and the result that the random Dirichlet series of finite order has almost surely(a.s.) no deficient functions is proved.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Unification and infinite series

Some infinite series are analysed on the basis of the hypergeometric function and integer structure and modular rings. The resulting generalized functions are compared with differentiation of the ‘mother’ series.     

On the coefficients of Neumann series of Bessel functions

Recently Pogány and Süli (Proc. Amer. Math. Soc. 137 (7) (2009) 2363-2368) derived a closed-form integral expression for Neumann series of Bessel functions. In this note we precisely characterize the class of functions α that generate the integral representation of a Neumann series of Bessel functions in the sense that the restriction αN_|=(αn) of a function α to the set N of all positive integers is the sequence of coefficients of the initial Neumann series.     

两种基本函数项级数系数公式注记

分析部分高等数学教材在推导幂级数和傅里叶级数的系数公式时存在的不足,并从函数逼近论的角度,就此问题提出改进思想和方法.     

Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients

For about thirty years, time series models with time-dependent coefficients have sometimes been considered as an alternative to models with constant coefficients or non-linear models. Analysis based on models with time-dependent models has long suffered from the absence of an asymptotic theory except in very special cases. The purpose of this paper is to provide such a theory without using a locally stationary spectral representation and time rescaling. We consider autoregressive-moving average (ARMA) models with time-dependent coefficients and a heteroscedastic innovation process. The coefficients and the innovation variance are deterministic functions of time which depend on a finite number of parameters. These parameters are estimated by maximising the Gaussian likelihood function. Deriving conditions for consistency and asymptotic normality and obtaining the asymptotic covariance matrix are done using some assumptions on the functions of time in order to attenuate non-stationarity, mild assumptions for the distribution of the innovations, and also a kind of mixing condition. Theorems from the theory of martingales and mixtingales are used. Some simulation results are given and both theoretical and practical examples are treated. Received 2004; Final version 23 December 2004