scholarly journals Approximation of classes of Poisson integrals by Fejer sums

2018 ◽  
Keyword(s):  
Complex Plane ◽  
Upper Bounds ◽  
Periodic Functions ◽  
Poisson Integrals

For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.

scholarly journals APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

2020 ◽  
Vol 8 (2) ◽  
pp. 114-121
Author(s):  
O. Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Partial Sums ◽  
Asymptotic Formulas ◽  
Periodic Functions ◽  
Arithmetic Means ◽  
Poisson Integrals ◽  
Fourier Sums ◽  
Uniformly Convergent

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.

Approximative properties of the three-harmonic Poisson integrals on the classes $W^{r}_{\beta}H^{\alpha}$.

2020 ◽  
Vol 17 (4) ◽  
pp. 538-548
Author(s):  
Ulyana Hrabova ◽  
Inna Kal'chuk ◽  
Leontii Filozof
Keyword(s):  
Upper Bounds ◽  
Approximation Of Functions ◽  
Poisson Integrals

We obtained the asymptotic equalities for the least upper bounds of the approximation of functions from the classes $W^{r}_{\beta}H^{\alpha}$ by three-harmonic Poisson integrals in the case $r+\alpha\leq 3$ in the uniform metric.

On certain expansions of the solutions of the general Lamé equation

1942 ◽  
Vol 38 (4) ◽  
pp. 364-367 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Recurrence Relations ◽  
Characteristic Exponent ◽  
Periodic Functions ◽  
The Real ◽  
Lamé Equation ◽  
Arbitrary Complex ◽  
Principal Value ◽  
Image Position

1. In this paper I shall deal with the solutions of the Lamé equationwhen n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the formwhere θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.It is easy to obtain the system of recurrence relationsfor the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for whichk′ being the principal value of (1−k2)½

Approximation of the classes W^{r}_{\beta}H^{\alpha} by three-harmonic Poisson integrals

2019 ◽  
Vol 16 (3) ◽  
pp. 357-371
Author(s):  
Inna Kal'chuk ◽  
Vasyl' Kravets ◽  
Ulyana Hrabova
Keyword(s):  
Upper Bounds ◽  
Poisson Integrals

We obtain asymptotic equalities for the least upper bounds of deviations of the three-harmonic Poisson integrals from functions of the classes W^{r}_{\beta}H^{\alpha} in a uniform metric in the case r>3, 0 <= \alpha < 1.

Optimal Recovery for Some Infinitely Differentiable Periodic Functions

2011 ◽  
Vol 282-283 ◽  
pp. 240-243
Author(s):  
Pei Xin Ye ◽  
Xue Hua Li
Keyword(s):  
Convergence Rate ◽  
Interpolation Method ◽  
Optimal Recovery ◽  
Upper Bounds ◽  
Optimal Order ◽  
Trigonometric Interpolation ◽  
Periodic Functions ◽  
Optimal Convergence Rate ◽  
Optimal Convergence ◽  
Intrinsic Error

We determine the optimal convergence rate of Dirichlet interpolating algorithm. Asymptotic inequalities are found for the upper bounds of approximation by trigonometric interpolation on the classes of convolutions of periodic functions admitting regular. The optimal order ofm-th minimum linear intrinsic error is determined. By means of previous results aboutn-widths, we discuss the optimality of interpolation method.

scholarly journals Asymptotics of approximation of conjugate functions by Poisson integrals

2019 ◽  
Vol 22 (2) ◽  
pp. 235-243
Author(s):  
Yu. I. Kharkevych ◽  
K. V. Pozharska
Keyword(s):  
Upper Bound ◽  
Arbitrary Order ◽  
Conjugate Functions ◽  
Periodic Functions ◽  
Poisson Integrals

We obtain a decomposition of the upper bound for the deviation of Poisson integrals of conjugate periodic functions. The decomposition enables one to provide the Kolmogorov–Nikol'skii constants of an arbitrary order.

On some higher order equations admitting meromorphic solutions in a given domain

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Grigor Barsegian ◽  
Fanning Meng
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Recent Trend ◽  
Upper Bounds ◽  
Higher Order ◽  
General Nature ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Similar Solutions ◽  
Higher Order Differential Equations

AbstractThis paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.

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