Partial Sums Of Fourier Series Research Articles

APPROXIMATION OF FUNCTIONS BY THE MATRIX MEANS IN WEIGHTED ORLICZ SPACES

Abstract. In this work, the approximation properties of the matrix submethods in weighted Orlicz spaces L T M , are investigated. We obtain some results related to trigonometric approximation using matrix submethods of partial sums of Fourier series of functions in weighted Orlicz spaces.The degree of trigonometric approximations by the matrix methods to the functions have been investigated in weighted Orlicz spaces. The error of estimations in this work is obtained in more general terms.

An asymptotic development of the Poisson integral for Laguerre polynomial expansions

The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in Lp spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions f∈Lp0,+∞ with 4/3<p<4. In this paper we deal with the Poisson integral Arf0<r<1 which arises by applying Abel’s summation method to the Laguerre expansion of the function f. About 50 years ago, Muckenhoupt intensively studied the Poisson integral for the Laguerre and Hermite polynomials. Among other things he proved pointwise convergence, the convergence by norm, and that the Poisson integral is a contraction mapping in Lp0,∞. Toczek and Wachnicki gave a Voronovskaja-type theorem by calculating the limit 1−r−1Arfx−fx as r→1−, provided that f′′x exists. We generalize this formula by deriving a complete asymptotic development. All its coefficients are explicitly given in a concise form. As an application we apply extrapolation methods in order to improve the rate of convergence of Arfx as r→1−.

On the Uniform Convergence of Spherical Partial Sums of Fourier Series by the Double Walsh System

On the Uniform Convergence of Spherical Partial Sums of Fourier Series by the Double Walsh System

Approximation of classes of Poisson integrals by Fejer means

The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series.&#x0D; The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials.&#x0D; In the paper, we study upper asymptotic estimates for deviations between a function and the Fejer means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane.An asymptotic equality for the upper bounds of Fejer means deviations on classes of Poisson integrals was obtained.

Approximation of classes of Poisson integrals by Fejer means

The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

Cesàro summability of subsequence of the partial sums of Fourier series in operator-valued setting

Cesàro summability of subsequence of the partial sums of Fourier series in operator-valued setting

Approximation Properties of the Vallée-Poussin Means Similar to the Partial Sums of Fourier Series in Laguerre–Sobolev Polynomials

Approximation Properties of the Vallée-Poussin Means Similar to the Partial Sums of Fourier Series in Laguerre–Sobolev Polynomials

Gibbs phenomena for some classical orthogonal polynomials

The Gibbs phenomena associated to partial sums of Fourier series are now well understood. In this paper, we show that Gibbs phenomena also occur for expansions of functions in terms of members of one of several general classes of orthogonal polynomials, in particular treating in a relatively unified manner expansions in either Legendre (more generally Jacobi), Hermite, or Laguerre polynomials. Noteworthy is the fact that the Gibbs constants associated to all of these expansions have the same value, approximately 0.0893 or more precisely 1π∫0πsin⁡ttdt−12.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

Calculation of fractional integrals using partial sums of Fourier series for structural mechanics problems

The goal of this study is to develop and apply an approximate method for calculating integrals that are part of models using Riemann-Liouville integrals, and to create a software product that allows such calculations for given functions. The main results of the study consist in the construction of a quadrature formula for an integral, and the cases where the density of the integral is a function from the spaces of continuous functions with generalized derivatives with weight and the Helder classes of functions with weight were considered. For the proposed quadrature formula we further investigated the error of its approximation in the spaces of continuous functions and quadratic-summing functions with weight. As a result of the study, effective error estimates of the approximating apparatus in the proposed classes of functions have been established. In addition, the approximated method has been implemented on the computer in the form of a program in the C language. The significance of the obtained results for the construction industry consists in the fact that when solving problems, including problems on finding the shapes of structures, taking into account the properties of materials, environmental changes, in the models of which the Riemann-Liouville integrals are used, it will be possible to apply an approximate approach, the quadrature formula proposed in the article.

О приближении средними Фейера классов аналитических периодических функций

Работа посвящена вопросам приближения периодических функций высокой гладкости средними арифметическими сумм Фурье. Наиболее естественным и простым примером линейного процесса аппроксимации периодических непрерывных функций действительной переменной является приближение елементами последовательности частичных сумм ряда Фурье. Известно, что последовательности частичных сумм ряда Фурье не являются равномерно сходящимися на всем пространстве $C$ $2\pi$-периодических непрерывных функций. Поэтому значительное число работ данного направления посвящено изучению аппроксимативных свойств других методов приближения, которые для заданной функции $f$ образуются с помощью некоторых преобразований частичных сумм ее ряда Фурье, и позволяют построить последовательности тригонометрических полиномов, которые равномерно сходятся для каждой функции $f \in C$. В частности, на протяжении последних десятилетий интенсивно изучаются суммы Валле Пуссена и суммы Фейера. В настоящее время в публикациях этой тематики накоплено значительное количество фактического материала. Одним из наиболее важных направлений в этой области является изучение асимптотического поведения верхних граней уклонений средних арифметических сумм Фурье по различным классам периодических функций. Методы исследования интегральных представлений уклонений тригонометрических полиномов, которые порождаются линейными методами суммирования рядов Фурье, возникли и получили свое развитие в работах С.М.~Никольского, С.Б.~Стечкина, Н.П.~Корнейчука, В.К. Дзядыка и их учеников.Целью работы является систематизация известных результатов, касающихся приближения классов периодических функций высокой гладкости средними арифметическими сумм Фурье, и представление новых фактов, полученных для их частных случаев.В работе изучено аппроксимативные свойства сумм Фейера на классах периодических функций, которые можно регулярно продолжить в соответствующую полосу комплексной плоскости. Получена асимптотическая формула для верхних граней уклонений в равномерной метрике сумм Фейера на классах интегралов Пуассона. Полученная формула обеспечивает решение соответствующей задачи Колмогорова-Никольского без дополнительных условий.

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

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 Chebyshev–Markov rational fractions are considered. Pointwise and uniform estimates for approximations are established. For the case in which the derivative of the measure is weakly equivalent to a power function, an asymptotic expression for the majorant of uniform approximations and an optimal parameter value ensuring the greatest rate of approximation by the method used in the paper are found. In the case of the even multiplicity of the poles of the approximating function, the asymptotic estimate is sharp. Examples of approximations of concrete functions are given.

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

Рассматриваются приближения на отрезке $[-1,1]$ функций, представляющих собой комбинацию классических функций Маркова, частичными суммами рядов Фурье по системе рациональных дробей Чебышева-Маркова. Устанавливаются поточечная и равномерная оценки приближений. В случае, когда производная меры слабо эквивалентна некоторой степенной функции, установлены асимптотическое выражение мажоранты равномерных приближений и оптимальное значение параметра, обеспечивающее наибольшую скорость приближений используемым методом. В случае четной кратности полюсов аппроксимирующей функции асимптотическая оценка точна. Приводятся примеры аппроксимаций индивидуальных функций. Библиография: 17 названий.

Transformation of the Mean Value of Integral On Fourier Series Expansion

Approximation of sigma is a damping factor which is obtained through transformation of mean value of integral to the functionality expanded via Fourier series. Where the result of the transformation is in the form of oscillation function, so as to form a modified partial sums of Fourier series. Through modification of the partial sums of Fourier series, the leap (overshoot) near discontinuity points of the oscillations function can be suppressed.

Modelling of reinforcement two-layer stitched woven fabric structure intended for composites

The main aim of this study is to establish a model for evaluating the internal geometry of two-layer stitched woven fabric structure. It starts with the basic description of the geometry of woven fabric and then discusses different geometric models and their limitations. Afterwards, it proceeds to the geometry of binding cell in plain weave for single-layer and two-layer stitched woven fabrics. A new concept of definition of central line of interlacing threads (binding wave) inside the fabric has been presented and then internal geometry of the fabric has been predicted by Fourier series (FS). It has also provided a mathematical analysis for the model validation by studying the approximation for two-layer plain stitched woven fabrics by a partial sum of FS, which is the proposed model. At the end, the influence of plain weave repeat size in binding wave of two-layer stitched woven cross-section has been studied. It has been found that with the increase in number of plain weaves in non-stitching section, the number of terms for binding wave required to get a better approximation also increases.

Definition of Set of diagnostic Parameters of System based on the Functional Spaces Theory

The development of methodical and mathematical apparatus for formation of a set of diagnostic parameters of complex technical systems, the content of which consists of processing the trajectories of the output processes of the system using the theory of functional spaces, is considered in this paper. The trajectories of the output variables are considered as Lebesgue measurable functions. It ensures a unified approach to obtaining diagnostic parameters regardless a physical nature of these variables and a set of their jump-like changes (finite discontinuities of trajectories). It adequately takes into account a complexity of the construction, a variety of physical principles and algorithms of systems operation. A structure of factor-spaces of measurable square Lebesgue integrable functions, ( spaces) is defined on sets of trajectories. The properties of these spaces allow to decompose the trajectories by the countable set of mutually orthogonal directions and represent them in the form of a convergent series. The choice of a set of diagnostic parameters as an ordered sequence of coefficients of decomposition of trajectories into partial sums of Fourier series is substantiated. The procedure of formation of a set of diagnostic parameters of the system, improved in comparison with the initial variants, when the trajectory is decomposed into a partial sum of Fourier series by an orthonormal Legendre basis, is presented. A method for the numerical determination of the power of such a set is proposed.&#x0D; New aspects of obtaining diagnostic information from the vibration processes of the system are revealed. A structure of spaces of continuous square Riemann integrable functions ( spaces) is defined on the sets of vibrotrajectories. Since they are subspaces in the afore mentioned factor-spaces, the general methodological bases for the transformation of vibrotrajectories remain unchanged. However, the algorithmic component of the choice of diagnostic parameters becomes more specific and observable. It is demonstrated by implementing a numerical procedure for decomposing vibrotrajectories by an orthogonal trigonometric basis, which is contained in spaces. The processing of the results of experimental studies of the vibration process and the setting on this basis of a subset of diagnostic parameters in one of the control points of the system is provided.&#x0D; The materials of the article are a contribution to the theory of obtaining information about the technical condition of complex systems. The applied value of the proposed development is a possibility of their use for the synthesis of algorithmic support of automated diagnostic tools.

On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Lebesgue space with variable exponent

In this paper we investigate some properties of approximation polynomials in particular de la Vallee-Poussin means, Fejer means and partial sums of Fourier series in weighted Lebesgue spaces with variable exponent. In addition to these we prove a simultaneous type theorem and some theorems on the equivalence of modulus of smoothness and the K-functional in weighted Lebesgue space with variable exponent.

Uniform convergence of orthogonal polynomial expansions for exponential weights

We consider an exponential weight $w(x) = \exp(-Q(x))$ on ${\mathbb R} = (-\infty,\infty)$, where $Q$ is an even and nonnegative function on ${\mathbb R}$. We always assume that $w$ belongs to a relevant class $\mathcal{F}(C^2+)$. Let $\{p_n\}$ be orthogonal polynomials for a weight $w$. For a function $f$ on ${\mathbb R}$, $s_n(f)$ denote the $(n-1)$-th partial sum of Fourier series. In this paper, we discuss uniformly convergence of $s_n(f)$ under the conditions that $f$ is continuous and has a bounded variation on any compact interval of ${\mathbb R}$. In the proof of main theorem, Nikolskii-type inequality and boundedness of the de la Vall{\'{e}}e Poussin mean of $f$ play important roles.

Approximation of signals by general matrix summability with effects of Gibbs Phenomenon

In the proposed paper the degree of approximation of signals (functions) belonging to $Lip(\alpha,p_{n})$ class has been obtained using general sub-matrix summability and a new theorem is established that generalizes the results of Mittal and Singh [10] (see [M. L. Mittal and Mradul Veer Singh, Approximation of signals (functions) by trigonometric polynomials in $L_{p}$-norm, \textit{Int. J. Math. Math. Sci.,} \textbf{2014} (2014), ArticleID 267383, 1-6 ]). Furthermore, as regards to the convergence of Fourier series of the signals, the effect of the Gibbs Phenomenon has been presented with a comparison among different means that are generated from sub-matrix summability mean together with the partial sum of Fourier series of the signals.

Modelling of multifilament woven fabric structure using Fourier series

Woven fabric structure and geometry has been described mathematically but not as good as it should be. In this study, an attempt is made to create a theoretical model on the geometry of plain single layer and two layer stitched woven structures and verify them with experimental results. The first part of the work deals with the model development and the second part reports on model validation. This part of the work starts with the basic description of the geometry of woven fabric. It then discusses different geometric models and their limitations. Afterwards, it proceeds to the geometry of binding cell in plain weave for single layer and two layer woven fabrics. At the end, it provides examples for the model validation by studying the approximation for plain woven fabrics by a partial sum of Fourier series. The approximation of the experimental spectrum holds good with the theoretical spectrum obtained by the model created.