Approximation Of Continuous Periodic Functions Research Articles

Asymptotic estimates for deviations of Fejér means on Poisson integrals

The work is devoted to the investigation of problem of approximation of continuous periodic functions of a real variable by trigonometric polynomials generated by linear methods of summation of Fourier series. The simplest example of the process of linear approximation of periodic functions is approximation of functions by partial sums of their Fourier series. However, sequences of partial Fourier sums are not uniformly convergent on the whole class of continuous periodic functions. Therefore, numerous studies are devoted to the study of approximation properties of approximating methods, which are generated by different transformations of sequences of partial sums of the Fourier series and allow obtaining sequences of trigonometric polynomials that are uniformly convergent for the entire class of continuous functions. In particular, Fejér means have been intensively studied in recent decades. One of the important tasks in this direction is study of the asymptotic behavior of the upper bounds of the deviations of trigonometric polynomials for a fixed class of periodic functions. The aim of the work is to systematize the known results concerning the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums, and to present new facts obtained for a more general case. In paper we investigate the asymptotic behavior of upper bounds on classes of Poisson integrals of periodic functions of real variable of deviations of linear means of Fourier series, which are constructed using the Fejér summation method. We study the classes consist of analytic functions of a real variable, which can be regularly extended into the corresponding strip of the complex plane. In the work, asymptotic inequalities for the upper bounds of the deviations of the Fejér means on the class of Poisson integrals were found. In certain case of parameters defining the class, the obtained formulas coincide with previously known asymptotic equalities.

Estimation of the Approximation of Continuous Periodic Functions by Fourier Sums

Estimation of the Approximation of Continuous Periodic Functions by Fourier Sums

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.
 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.
 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.

Power of Dirichlet kernels and approximation by discrete linear operators $\textit{I}$: direct results

The second and third powers of the Dirichlet kernel are used to construct discrete linear operators for the approximation of continuous periodic functions. An estimate of the rate of convergence is given. Approximation of non-periodic functions are also considered.

On multidimensional Jackson's theorem

We have found the best linear polynomial methods of approximation of continuous periodic functions of multiple variables in uniform metric with concave modulus of continuity.

On one method of approximation of continuous periodic functions by trigonometric polynomials

On one method of approximation of continuous periodic functions by trigonometric polynomials.

On mean approximation of continuous periodic functions by functions that satisfy Lipschitz condition

On mean approximation of continuous periodic functions by functions that satisfy Lipschitz condition.

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.

Asymptotics of approximation of continuous periodic functions by linear means of their Fourier series

We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error , . This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.

APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

Almost Coconvex Approximation of Continuous Periodic Functions

If a 2 𝜋 -periodic function f continuous on the real axis changes its convexity at 2s, s ∈ ℕ, inflection points yi : − 𝜋 ≤ y2s< y2s−1< ... < y1< 𝜋 and, for all other i ∈ ℤ, yi are periodically defined, then, for any natural n ≥ Nyi, we can find a trigonometric polynomial Pn of order cn such that Pn has the same convexity as f everywhere except, possibly, small neighborhoods of the points yi : (yi− 𝜋/n, yi + 𝜋/n) and, moreover,$$ \left\Vert f-{P}_n\right\Vert \le c(s){\omega}_4\left(f,\uppi /n\right), $$ where Nyi is a constant that depends only on mini=1,...,2s{yi− yi+1}, c and c(s) are constants thatdepend only on s, 𝜔4(f, ·) is the fourth modulus of smoothness of the function f, and ‖⋅‖is the max-norm.

Approximation of Continuous Periodic Functions of Two Variables via Power Series Methods of Summability

We prove a Korovkin type approximation theorem via power series methods of summability for continuous $2\pi$-periodic functions of two variables and verify the convergence of approximating double sequences of positive linear operators by using modulus of continuity. An example concerning double Fourier series is also constructed to illustrate the obtained results.

On approximation of continuous periodic functions by Stechkin's polynomials

The answer to the V.I. Ivanov's question about the exact order approximation by Stechkin’s polynomials where s = 3 and s = 4 is obtained (s is the order of the modulus of smoothness). We use the method of Fourier multipliers and special differencing operators. The case where s = 2 was studied previously.

Lower bounds for the deviations of the best linear methods of approximation of continuous functions by trigonometric polynomials

In the case of uniform approximation of continuous periodic functions of one variable by trigonometric polynomials, we obtain lower bounds for the Jackson constants of the best linear methods of approximation.

Integral approximation of the characteristic function of an interval and the Jackson inequality in C( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFtcpvaaa!46CC! $$ \mathbb{T} $$ )

We apply the results on the integral approximation of the characteristic function of an interval by the subspace \( \mathcal{T}_{n - 1} \) of trigonometric polynomials of order at most n − 1, which were obtained by the authors earlier, to investigate the Jackson inequality between the best uniform approximation of a continuous periodic function by the subspace \( \mathcal{T}_{n - 1} \) and its modulus of continuity of the second order. The corresponding method of the uniform approximation of continuous periodic functions by trigonometric polynomials is constructed.

Approximation of Continuous Periodic Functions Via Statistical Convergence

In this work, we give a nontrivial generalization of the classical Korovkin approximation theorem by using the concept of A-statistical convergence, which is a regular (nonmatrix) summability method, for sequences of positive linear operators denned on the space of all real-valued continuous and 2π periodic functions on the real m-dimensional space. Furthermore, in the case of m = 2, we display an application which shows that our result is stronger than the classical approximation.

Directional K- Functionals Claudia Cottin

When considering approximation of continuous periodic functions f: Rd → R by blending-type approximants which depend on directions ξ1,…,ξν ∈ Rd directional moduli of smoothness (1) are appropriate measures of smoothness of /. In this paper, we introduce equivalent directional K- functionals. As an application, we obtain a result on the degree of approximation by certain trigonometric blending functions.

Asymptotic estimates of approximation of continuous periodic functions by Fourier sums

Asymptotic estimates, expressed in terms of the value of the modulus of continuity of r-th order (r≥2) at the point t=π/n of a functionfeC2π or of the (ψ, β)-derivative of a functionf∈CBψC, are established for the deviations of continuous periodic functions from their Fourier sums.

Approximation of continuous periodic functions by Bernshtein sums

Approximation of continuous periodic functions by Bernshtein sums