On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

Mathematical Notes - Approximations on the closed interval $$[-1,1]$$ of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of...     

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite-dimensional polydisk, as well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on the infinite-dimensional unit ball. We use both techniques from the theory of sampling in Paley-Wiener spaces, and classical results from analytic number theory.     

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.     

A relaxed estimate of the degree of approximation by Fourier series in generalized Hölder metric

The aim of the paper is to give a relaxed estimate pertaining to the degree of approximation of the partial sums of Fourier series in a new Banach space of functions introduced by Das, Nath and Ray [2]. Furthermore, applying our new result, we verify, under certain natural conditions, that some classical means have the same approximation degree as the partial sums.     

一类随机变量序列加权和的Chover型重对数律

对于具有某种尾渐近行为的独立同分布的随机变量序列,本文通过积分检验刻划了其加权部分和的极限结果,并作为推论获得了Chover型重对数律。把这些结果应用到经典的可和方式,获得了相应的结果。     

A generalization of Jentzsch’s theorem

We study the asymptotic behavior of the roots of polynomials given by a linear summation method for partial sums of the Fourier series.     

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded almost surely. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type.     

Special series in Laguerre polynomials and their approximation properties

We consider special series in the classical Laguerre polynomials coinciding in particular cases with the mixed series associated to Laguerre polynomials, introduced by the author previously, as well as Fourier–Sobolev series in Sobolev orthogonal Laguerre polynomials. We address the questions of uniform convergence of these series on a finite segment of the positive half-axis. We study the approximation properties of partial sums of special series on the positive half-axis, with particular attention paid to estimating their Lebesgue function.     

Pointwise Behavior of Double Fourier Series of Functions of Bounded Variation

We extend some recent results of S. A. Telyakovskii on the uniform boundedness of the partial sums of Fourier series of functions of bounded variation to periodic functions of two variables, which are of bounded variation in the sense of Hardy. As corollaries, we obtain the classical Parseval formula, the convergence theorem of the series involving the sine Fourier coefficients, and a lower estimate of the best approximation by trigonometric polynomials in the metric of L in a sharpened version.     

On linear processes with dependent innovations

We consider asymptotic behavior of partial sums and sample covariances for linear processes whose innovations are dependent. Central limit theorems and invariance principles are established under fairly mild conditions. Our results go beyond earlier ones by allowing a quite wide class of innovations which includes many important nonlinear time series models. Applications to linear processes with GARCH innovations and other nonlinear time series models are discussed.     

On uniform approximation by partial sums of Jacobi series on elliptic region

In this paper, we discuss the relation between the partial sums of Jacobi series on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.     

Generation of symmetric exponential sums

In this paper, a new method for generation of infinite series of symmetric identities written for exponential sums in real numbers is proposed. Such systems have numerous applications in theory of numbers, chaos theory, algorithmic complexity, dynamic systems, etc. Properties of generated identities are studied. Relations of the introduced method for generation of symmetric exponential sums to the Morse-Hedlund sequence and to the theory of magic squares are established.     

Some generalizations of uniqueness theorems for series in Walsh systems

We determine conditions on the behavior of the partial sums of a Walsh series outside a certain U-set, under which this series is a Fourier-Walsh series.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 367–372, March, 1973.     

Limit Curves for Zeros of Sections of Exponential Integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\) , and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.     

Classical symmetric functions in superspace

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.     

Minkowski’s inequality and sums of squares

Positive polynomials arising from Muirhead’s inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski’s inequality can be rewritten as sums of squares.     

Optimal sequencing of a set of positive numbers with the variance of the sequence’s partial sums maximized

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.     

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.     

H_1(D)空间的Bessel级数

本文给出关于 Ｈ１（Ｄ）空间中函数的 Ｂｅｓｓｅｌ级数的部分和用幂级数的部分和表示的一个恒等式．基于它,可以得到Ｂｅｓｓｅｌ级数部分和偏差的诸多精确估计．     

H1（D）空间的Bessel级数

本文给出关于H1（D）空间中函数的Bessel级数的部分和用幂级的部分和表示的一个恒等式，基于它，可以得到Bessel 级数部分和偏差的诸多精确估计。