n元等比级数

定义了n元等比数列、n元等比级数,给出了它们的通项公式及前n项和.并解决了多元等比级数的敛散性问题,求出了多元等比级数的和.指出了多元等比数列及多元等比级数在特殊情况下与一元等比数列及一元等比级数的一致性.     

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.     

On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-christoffel rules, I

Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval. Gauss quadrature rules are designed for each of them. Various questions concerning their construction and application are studied, theoretically or experimentally. They are so efficient that they should be considered for the development of software for special functions. Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.     

ON ORDER AND TYPE OF MULTIPLE DIRICHLET SERIES

This article investigates the growth of multiple Dirichlet series.The order and the type of n-tuple Dirichlet series in C~n are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained,which generalize some results about simple Dirichlet series of Lindelof and Pringsheim.     

基于结构元线性生成的复Fuzzy值函数项级数

文献[1]中提出了基于结构元理论的Fuzzy数项级数的概念,文献[2]、文献[3]、文献[4]对其收敛性进行了探讨,文献[5]、文献[6]对模糊值函数项数列及级数进行了研究。本文在此基础上给出了基于结构元线性生成的复Fuzzy值函数项数列及级数的定义,同时对复Fuzzy值函数项级数的一些重要性质进行了研究,并给出了相应定理。     

Self-criticality and stochasticity of an S&P 500 index time series

In this paper, three time series representative of the daily high, low and closing prices of S&P 500 index time series, as from 1 December 1988 to 1 April 1998 are studied. The hypothesis advanced by Osborne that the stock market time series satisfy a log-normal distribution is rejected. The self-critical behavior of these time series is investigated. A fractional Brownian motion model for such time series is supported. Arguments are directed torwards a negation of a chaotic explanation of these time series.     

On the generation of high-quality white noise series

A new approach to the simulation of white-noise time-series is presented. The approach is based on the frequency-domain property of white-noise as having a flat power spectrum density (PSD). From such a PSD, a linear complex spectrum of random-phase multisinusoidal series (MS) may be generated. Next, the fast Fourier transform is applied to this linear spectrum to generate a random-phase multisinusoidal N-sample series, simulating the white-noise series. The properties of some MS series are discussed including a gaussian white noise multisinusoidal series. The idea is illustrated by a number of examples. They demonstrate that the spectral and correlation properties of such series are enhanced in comparison to the properties of the random phases used to generate them. They also demonstrate that the spectral and correlation properties of such series are better, especially for short series, in comparison with standard white noise generators.     

在全平面上解析的Laplace-Stieltjes变换的正规增长性

本文研究了应用Knopp-Kojima的方法得到的Dirichlet级数正规增长性.利用Dirichlet级数和Laplace-Stieltjes变换的关系,得到了Laplace-Stieltjes变换在全平面上的正规增长级的等价条件.     

简化原理在Hilbert空间与可分Banach空间的一些应用

本文研究了简化原理在Hilbert空间与可分Banach空间中的一些应用,利用简化原理和独立随机元收敛准则获得了中分Banach空间随机级数的收缩原理和B-值随机Dirichlet级数简单收敛横坐标及一般随机整函数的增长性和值分布,将许多以Rademacher序列为系数的随机Tayor级数和随机Dirichlet级数的相关结果,推广到一般的具有独立对称分布系数的随机级数上去。     

Numeration of non-decreasing and non-increasing serial sequences

Finite sets of n-valued serial sequences are examined. Their structure is determined not only by restrictions on the number of series and series lengths, but also by restrictions on the series heights, which define the order number of series and their lengths, but also is limited to the series heights, by whose limitations the order of series of different heights is given. Solutions to numeration and generation problems are obtained for the following sets of sequences: non-decreasing and non-increasing sequences where the difference in heights of the neighboring series is either not smaller than a certain value or not greater than a certain value. Algorithms that assign smaller numbers to lexicographically lower sequences and smaller numbers to lexicographically higher sequences are developed.     

Trigonometric sums and fresnel-type integrals

If the partial sums of a trigonometric series are non-negative and two additional conditions are satisfied, then the given series is a Fourier series. One of these conditions is analysed here and necessary expansions and numerical values are given.     

随机Dirichlet级数的增长性(Ⅱ)

虽然有许多关于半平面上收敛的Dirichlet级数和随机Dirichlet级数增长性的文章,但对零级的随机Dirichlet级数没有满足的结果,本文研究了零级的随机Dirichlet级数的增长性,并得到一些充要条件。     

Note on the differentiability of arithmetic Fourier series arising from Eisenstein series

The aim of this note is to present results concerning the differentiability of some Fourier series arising from Eisenstein series. Sine series exhibit different behaviours with respect to differentiability than the series with cosine function. The precise results are given for the series related to Eisenstein series of weight 2, whereas for the series arising from Eisenstein series of higher weight we conjecture the results.     

Nonparametric estimation of the spectral density of amplitude-modulated time series with missing observations

Consider a real-valued and second-order stationary time series with mean zero. The aim is to estimate its spectral density. A minimax solution of this problem is known when either the time series is observed directly, or some observations are missed according to an independent Bernoulli process, or for some special cases when the time series is multiplied by an amplitude-modulating time series with known distribution. It is shown that if a time series of interest, a Bernoulli time series defining missing mechanism, and an amplitude-modulating time series are mutually independent, then the shape of spectral density of an underlying time series of interest can be estimated with the minimax rate known for the case of direct observations. Furthermore, in some special cases the spectral density can be estimated with the minimax rate known for directly observed time series of interest.     

平面上零级Dirichlet级数和随机Dirichlet级数的上下级

本文对平面上的零级Dirichlet级数和随机Dirichlet级数的上下级进行了研究.利用Newton多边形,在一定条件下得到了Dirichlet级数和随机Dirichlet级数的上下级与其系数的重要关系.推广了平面上的零级Dirichlet级数和随机Dirichlet级数的增长性的研究范围.     

On the series of absolute values of blocks of trigonometric series

Conditions of integrability with a power weight are obtained for the sums of series of absolute values of blocks of sine series and cosine series under certain special conditions imposed on the coefficients of these series.     

The lower order and linear order of multiple dirichlet series

The article investigates the growth of multiple Dirichlet series.The lower order and the linear order of n-tuple Dirichlet series in C~n are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained.     

On a problem of littlewood

The theorem on the tending to zero of coefficients of a trigonometric series is proved when theL ¹-norms of partial sums of this series are bounded. It is shown that the analog of Helson's theorem does not hold for orthogonal series with respect to the bounded orthonormal system. Two facts are given that are similar to Weis' theorem on the existence of a trigonometric series which is not a Fourier series and whoseL ¹-norms of partial sums are bounded.     

Clustering Nonlinear,Nonstationary Time Series Using BSLEX

Accurate clustering of time series is a challenging problem for data arising from areas such as financial markets, biomedical studies, and environmental sciences, especially when some, or all, of the series exhibit nonlinearity and nonstationarity. When a subset of the series exhibits nonlinear characteristics, frequency domain clustering methods based on higher-order spectral properties, such as the bispectra or trispectra are useful. While these methods address nonlinearity, they rely on the assumption of series stationarity. We propose the Bispectral Smooth Localized Complex EXponential (BSLEX) approach for clustering nonlinear and nonstationary time series. BSLEX is an extension of the SLEX approach for linear, nonstationary series, and overcomes the challenges of both nonlinearity and nonstationarity through smooth partitions of the nonstationary time series into stationary subsets in a dyadic fashion. The performance of the BSLEX approach is illustrated via simulation where several nonstationary or nonlinear time series are clustered, as well as via accurate clustering of the records of 16 seismic events, eight of which are earthquakes and eight are explosions. We illustrate the utility of the approach by clustering S&P 100 financial returns.     

Martingales and some integrability criterions for multiple trigonometric series

Applying some martingale properties of the sequences of cubic partial sums of series in the Haar multiple system, some integrability criterions for multiple trigonometric series are established. Some of the proved theorems improve and extend to the multiple trigonometric series the classical theorems on integrability of one-dimensional trigonometric series, the proofs of which are based on Helly??s principal of choice.