Next Article in Journal
On the Bias in Confirmatory Factor Analysis When Treating Discrete Variables as Ordinal Instead of Continuous
Next Article in Special Issue
BMO and Asymptotic Homogeneity
Previous Article in Journal
Data Analysis Using a Coupled System of Ornstein–Uhlenbeck Equations Driven by Lévy Processes
Previous Article in Special Issue
Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes

by
Inna Kal’chuk
and
Yurii Kharkevych
*
Faculty of Information Technologies and Mathematics, Lesya Ukrainka Volyn National University, 43025 Lutsk, Ukraine
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(4), 161; https://doi.org/10.3390/axioms11040161
Submission received: 28 February 2022 / Revised: 26 March 2022 / Accepted: 28 March 2022 / Published: 1 April 2022
(This article belongs to the Special Issue Approximation Theory and Related Applications)

## Abstract

:
Asymptotic equalities are obtained for the least upper bounds of approximations of functions from the classes $W β , ∞ r$ by the generalized Abel-Poisson integrals $P γ ( δ ) , 0 < γ ≤ 2 ,$ for the case $r > γ$ in the uniform metric, which provide the solution to the Kolmogorov–Nikol’skii problem for the given method of approximation on the Weyl-Nagy classes.
MSC:
42A05; 41A60

## 1. Introduction

Let L be a space of $2 π$-periodic summable functions and
$S [ f ] = a 0 2 + ∑ k = 1 ∞ ( a k cos k x + b k sin k x )$
be the Fourier series of $f ∈ L$.
Further, let C be a subset of the continuous functions from L with the uniform norm $∥ f ∥ C = max t | f ( t ) | ;$ $L ∞$ be a subset of the functions $f ∈ L$ with the finite norm $∥ f ∥ ∞ = ess sup t | f ( t ) | .$
Let $Λ = { λ δ ( k ) }$ be the set of functions depending on $k ∈ N ∪ 0$ and on the parameter $δ ∈ E Λ ⊂ R$, the set $E Λ$ has at least one limit point and $λ δ ( 0 ) = 1$. Using the set $Λ$ to each function $f ∈ L$ we can associate the series
$a 0 2 + ∑ k = 1 ∞ λ δ ( k ) ( a k cos k x + b k sin k x ) , δ ∈ E Λ ,$
which converges for every $δ ∈ E Λ$ and all x to the continuous function $U δ ( f ; x ; Λ )$.
If the series
$1 2 + ∑ k = 1 ∞ λ δ ( k ) cos k t$
is the Fourier series of some summable function, then (similarly to ([1], p. 52)) for almost all $x ∈ R$ we have the equality
$U δ ( f ; x ; Λ ) = 1 π ∫ − π π f ( x + t ) 1 2 + ∑ k = 1 ∞ λ δ ( k ) cos k t d t .$
Putting in the equality (1) $λ δ ( k ) = e − k γ δ , 0 < γ ≤ 2 ,$ we obtain the quantity
$U δ ( f ; x ; Λ ) : = P γ ( δ ; f ; x ) = 1 π ∫ − π π f ( x + t ) 1 2 + ∑ k = 1 ∞ e − k γ δ cos k t d t , δ > 0 , 0 < γ ≤ 2 ,$
which is usually called the generalized Abel-Poisson integral of the function f (see, e.g., [2,3]). For $γ = 1$ the integral (2) is the Poisson integral (see, e.g., [4]), for $γ = 2$ the integral (2) is the Weierstrass integral (see, e.g., [5]).
Let us define the classes of functions that we consider further. Let $f ∈ L$, $r > 0$ and $β$ be a real number. If the series
$∑ k = 1 ∞ k r a k cos k x + β π 2 + b k sin k x + β π 2$
is the Fourier series of a summable function, then it is denoted by $f β r$ and is called the $r , β$-derivative of the function f in the Weyl-Nagy sense (see, e.g., [6]). Let $W β , ∞ r$ be the classes of the functions f for which $∥ f β r ( · ) ∥ ∞ ≤ 1$.
In this paper, we consider the problem of asymptotic behavior as $δ → ∞$ of the quantity
$E ( W β , ∞ r ; P γ ( δ ) ) C = sup f ∈ W β , ∞ r ∥ f ( · ) − P γ ( δ , f , · ) ∥ C .$
If the function $g ( δ )$ is found in an explicit form such that
$E ( W β , ∞ r ; P γ ( δ ) ) C = g δ + o g δ , δ → ∞ ,$
then according to Stepanets [6] we say that the Kolmogorov–Nikol’skii problem is solved for the class $W β , ∞ r$ and the generalized Abel-Poisson integral in the uniform metric.
The approximation properties of the generalized Poisson integrals have been studied only in the cases $γ = 1$ (Poisson integral) and $γ = 1$ (Weierstrass integral). In particular, the Kolmogorov–Nikol’skii problems for the Poisson integral on the different functional classes have been solved in [7,8,9,10,11]. Similar problems for Weierstrass integral have been solved in [5,12,13,14].
Regarding the results of estimating the approximation rate by the generalized Poisson integrals for $0 < γ ≤ 2$ we note the work [2], where the approximation properties of the integrals (2) on Zygmund classes $Z α , 0 ≤ α ≤ 2 ,$ have been studied.
In this paper, we aim to find asymptotic equations for quantities (3) for arbitrary $0 < γ ≤ 2$. This will allow us to find such $γ$ for any r, so that the approximation rate of functions from the classes $W β , ∞ r$ by the generalized Abel-Poisson integrals, i.e, the rate at which the quantity (3) tends to zero, is equal to $1 δ$. This approximation rate could not be achieved when approximating by Poisson integrals and Weierstrass integrals.
At present, the extremal problems of the approximation theory, being related to the study of the approximation properties of linear methods for summing Fourier series, become increasingly relevant in applied mathematics, in particular, in the creation of mathematical models [15,16,17,18,19], in signal transmission [20,21], in the decision theory [22] and others. The problem considered in the paper, as well as those close to it [23,24,25] find practical application in the issues of coding, transmission and reproduction of images.

## 2. Main Result

Let us define the summing function for the generalized Abel-Poisson integral as follows
$τ ( u ) = 1 − e − u γ ( γ − r − 1 ) δ r + 2 γ − 1 u 2 − γ + ( 2 + r − γ ) δ r + 1 γ − 1 u 1 − γ , 0 ≤ u ≤ 1 δ γ , 1 − e − u γ u − r , u ≥ 1 δ γ ,$
where $0 < γ ≤ 2$, $δ > 0 .$
Theorem 1.
Let $r > γ$. Then the following asymptotic equality holds as $δ → ∞$:
$E W β , ∞ r ; P γ ( δ ) C = 1 δ sup f ∈ W β , ∞ r f 0 γ ( · ) C + O ( Υ ( δ , r , γ ) ) ,$
where $f 0 γ ( x )$ is $r , β$—derivative in the Weyl-Nagy sense as $r = γ , β = 0$ and
$Υ ( δ , r , γ ) = 1 ( δ γ ) r , γ < r < 2 γ , ln δ δ 2 , r = 2 γ , 1 δ 2 , r > 2 γ .$
Proof.
Let us rewrite the function $τ ( u )$ given by (4) in the form $τ ( u ) = φ ( u ) + μ ( u )$ (see, e.g., [26]), where
$φ ( u ) = ( γ − r − 1 ) δ r + 2 γ − 1 u 2 + ( 2 + r − γ ) δ r + 1 γ − 1 u , 0 ≤ u ≤ 1 δ γ , u γ − r , u ≥ 1 δ γ .$
$μ ( u ) = = 1 − e − u γ − u γ ( γ − r − 1 ) δ r + 2 γ − 1 u 2 − γ + ( 2 + r − γ ) δ r + 1 γ − 1 u 1 − γ , 0 ≤ u ≤ 1 δ γ , 1 − e − u γ − u γ u − r , u ≥ 1 δ γ ,$
Further we show a summability of the transformations of the form
$φ ^ β ( t ) = φ ^ ( t , β ) = 1 π ∫ 0 ∞ φ ( u ) cos u t + β π 2 d u ,$
$μ ^ β ( t ) = μ ^ ( t , β ) = 1 π ∫ 0 ∞ μ ( u ) cos u t + β π 2 d u .$
First, prove a convergence of the integral
$A φ = 1 π ∫ − ∞ ∞ φ ^ β ( t ) d t .$
Integrating twice by parts and taking into account that $φ ( 0 ) = 0$, $lim u → ∞ φ ( u ) = lim u → ∞ φ ′ ( u ) = 0$ and $φ ′ ( u )$ is continuous on $0 , ∞$, we have
$∫ 0 ∞ φ ( u ) cos u t + β π 2 d u = 1 t 2 φ ′ ( 0 ) cos β π 2 − ∫ 0 ∞ φ ″ ( u ) cos u t + β π 2 d u .$
In view of the fact, that the function $φ ( u )$ is downward closed on $1 δ γ , ∞$, the last relation yields
$∫ 0 ∞ φ ( u ) cos u t + β π 2 d u ≤ 1 t 2 φ ′ ( 0 ) + ∫ 0 1 δ γ + ∫ 1 δ γ ∞ φ ″ ( u ) d u = = 1 t 2 ( 2 + r − γ ) δ r + 1 γ − 1 + 2 ( r + 1 − γ ) δ r + 1 γ − 1 + ∫ 1 δ γ ∞ φ ″ ( u ) d u = = 1 t 2 K 1 δ r + 1 γ − 1 − φ ′ 1 δ γ = K 2 δ r + 1 γ − 1 1 t 2 .$
Here and below we denote by symbols $K i , i = 1 , 2 , … ,$ some positive constants.
From the inequalities (10) it follows that
$∫ | t | ≥ δ γ ∫ 0 ∞ φ ( u ) cos u t + β π 2 d u d t = O δ r γ − 1 , δ → ∞ .$
By virtue of the equality (4.16) from ([1], p. 69), we obtain
$∫ 0 δ γ | ∫ 0 ∞ φ ( u ) cos u t + β π 2 d u | d t = ∫ 0 δ γ | ∫ 0 1 δ γ + ∫ 1 δ γ ∞ φ ( u ) cos u t + β π 2 d u | d t ≤ ≤ δ γ ∫ 0 1 δ γ φ ( u ) d u + ∫ 0 δ γ ∫ 1 δ γ 1 δ γ + 2 π t φ ( u ) d u d t ≤ K 3 δ r γ − 1 + ∫ 0 δ γ ∫ 1 δ γ 1 δ γ + 2 π t u γ − r d u d t .$
Making a change of variables and integrating by parts in the last integral, we obtain
$∫ 0 δ γ ∫ 1 δ γ 1 δ γ + 2 π t u γ − r d u d t = 2 π ∫ 2 π δ γ ∞ ∫ 1 δ γ 1 δ γ + x u γ − r d u d x x 2 = = 2 π − 1 x ∫ 1 δ γ 1 δ γ + x u γ − r d u | 2 π δ γ ∞ + ∫ 2 π δ γ ∞ 1 x 1 δ γ + x γ − r d x = = 2 π ( − lim x → ∞ 1 x ∫ 1 δ γ 1 δ γ + x u γ − r d u + δ γ 2 π ∫ 1 δ γ ( 1 + 2 π ) δ γ u γ − r d u + + δ r γ − 1 ∫ 2 π δ γ ∞ 1 x 1 + δ γ x γ − r d x ) .$
In view of
$lim x → ∞ 1 x ∫ 1 δ γ 1 δ γ + x u γ − r d u = 0 ,$
$δ γ 2 π ∫ 1 δ γ ( 1 + 2 π ) δ γ u γ − r d u = K 4 δ r γ − 1 ,$
$δ r γ − 1 ∫ 2 π δ γ ∞ 1 x 1 + δ γ x γ − r d x = δ r γ − 1 ∫ 1 + 2 π ∞ y γ − r y − 1 d y =$
$= δ r γ − 1 ∫ 1 + 2 π ∞ y γ − r − 1 1 + 1 y − 1 d y ≤ 1 + 1 2 π δ r γ − 1 ∫ 1 + 2 π ∞ y γ − r − 1 d y = K 5 δ r γ − 1 ,$
from (13) and (12) we can write that
$∫ 0 δ γ ∫ 0 ∞ φ ( u ) cos u t + β π 2 d u d t = O δ r γ − 1 , δ → ∞ .$
One can analogously show that
$∫ − δ γ 0 ∫ 0 ∞ φ ( u ) cos u t + β π 2 d u d t = O δ r γ − 1 , δ → ∞ .$
From the Formulas (11), (14) and (15) we obtain
$A ( φ ) = O δ r γ − 1 , δ → ∞ .$
Now we show the convergence of the integral
$A ( μ ) = 1 π ∫ − ∞ ∞ μ ^ β ( t ) d t .$
Integrating twice by parts and taking into account that $μ ( 0 ) = μ ′ ( 0 ) = 0 ,$ $lim u → ∞ μ ( u ) = lim u → ∞ μ ′ ( u ) = 0$, we have
$∫ 0 ∞ μ ( u ) cos u t + β π 2 d u = − 1 t 2 ∫ 0 ∞ μ ″ ( u ) cos u t + β π 2 d u ,$
and hence
$∫ 0 ∞ μ ( u ) cos u t + β π 2 d u ≤ 1 t 2 ∫ 0 ∞ μ ″ ( u ) d u = 1 t 2 ∫ 0 1 δ γ + ∫ 1 δ γ 1 + ∫ 1 ∞ μ ″ ( u ) d u .$
Further we use the notations
$V ( u ) = 1 − e − u γ − u γ u 2 − γ , W ( u ) = 1 − e − u γ − u γ u 1 − γ .$
Let us differentiate twice the functions $V ( u )$ and $W ( u )$:
$V ′ ( u ) = γ u ( e − u γ − 1 ) + ( 2 − γ ) u 1 − γ 1 − e − γ − u γ ,$
$W ′ u = γ ( e − u γ − 1 ) + ( 1 − γ ) u − γ 1 − e − u γ − u γ ,$
$V ″ ( u ) = γ e − u γ 1 − γ u γ − 1 + 2 − γ ( 1 − γ ) u − γ 1 − e − u γ − u γ + γ ( e − u γ − 1 ) ,$
$W ″ ( u ) = − γ 2 u γ − 1 e − u γ − γ γ − 1 u − γ − 1 1 − e − u γ − u γ + γ ( γ − 1 ) u − 1 e − u γ − 1 .$
By virtue of the fact, that for $u ∈ 0 , 1 δ γ$
$μ ″ ( u ) = ( γ − r − 1 ) δ r + 2 γ − 1 V ″ ( u ) + ( 2 + r − γ ) δ r + 1 γ − 1 W ″ ( u ) ,$
we obtain
$∫ 0 1 δ γ μ ″ ( u ) d u ≤ ( r + 1 − γ ) δ r + 2 γ − 1 ∫ 0 1 δ γ V ″ ( u ) d u + ( 2 + r − γ ) δ r + 1 γ − 1 ∫ 0 1 δ γ W ″ ( u ) d u .$
Taking into account, that for $u ∈ 0 , 1 δ γ$$V ″ ( u ) ≤ 0$, $W ″ ( u ) ≤ 0$, and also the inequalities
$1 − e − u γ ≤ u γ , e − u γ + u γ − 1 ≤ u 2 γ 2 ,$
we have
$∫ 0 1 δ γ μ ″ ( u ) d u ≤ ( r + 1 − γ ) δ r + 2 γ − 1 V ′ 0 − V ′ 1 δ γ + + ( 2 + r − γ ) δ r + 1 γ − 1 W ′ 0 − W ′ 1 δ γ = = ( r + 1 − γ ) δ r + 2 γ − 1 γ δ γ 1 − e − 1 δ + 2 − γ δ γ 1 − γ e − 1 δ + 1 δ − 1 + + ( 2 + r − γ ) δ r + 1 γ − 1 γ 1 − e − 1 δ + 1 − γ δ γ − γ e − 1 δ + 1 δ − 1 ≤ K 6 δ r + 1 γ − 2 .$
Noting, that for $u ≥ 1 δ γ$
$μ ″ ( u ) = r ( r + 1 ) 1 − e − u γ − u γ u − r − 2 − 2 γ u γ − r − 2 e − u γ − 1 + + γ ( γ − 1 ) u γ − 2 ( e − u γ − 1 ) − γ u 2 γ − 2 e − u γ u − r ,$
we can write
$∫ 1 δ γ 1 μ ″ ( u ) d u ≤ r ( r + 1 ) ∫ 1 δ γ 1 ( e − u γ + u γ − 1 ) u − r − 2 d u + + 2 γ r ∫ 1 δ γ 1 1 − e − u γ u γ − r − 2 d u + γ ∫ 1 δ γ 1 ( γ − 1 ) u γ − 2 ( e − u γ − 1 ) − γ u 2 γ − 2 e − u γ u − r d u .$
The inequality (19) in combination with
$( γ − 1 ) u γ − 2 ( e − u γ − 1 ) − γ u 2 γ − 2 e − u γ ≤ 2 γ − 1 u 2 γ − 2 , u ∈ 0 , ∞ ,$
yields
$∫ 1 δ γ 1 | μ ″ ( u ) | d u ≤ r ( r + 1 ) 2 + 2 γ r + γ ( 2 γ − 1 ) ∫ 1 δ γ 1 u 2 γ − r − 2 d u ≤ ≤ r ( r + 1 ) 2 + 2 γ r + γ ( 2 γ − 1 ) δ γ ∫ 1 δ γ 1 u 2 γ − r − 1 d u = K 7 δ 1 γ + K 8 δ r + 1 γ − 2 , r ≠ 2 γ , K 9 δ 1 γ ln δ , r = 2 γ ,$
In the case $u ∈ 1 , ∞$ we obtain
$∫ 1 ∞ μ ″ ( u ) d u ≤ r ( r + 1 ) ∫ 1 ∞ ( e − u γ + u γ − 1 ) u − r − 2 d u + + 2 γ r ∫ 1 ∞ 1 − e − u γ u γ − r − 2 d u + γ ∫ 1 ∞ ( γ − 1 ) u γ − 2 ( e − u γ − 1 ) − γ u 2 γ − 2 e − u γ u − r d u .$
Let $0 < γ < 1$, then using the inequalities (19) and (21), we obtain
$∫ 1 ∞ | μ ″ ( u ) | d u ≤ r ( r + 1 ) 2 + 2 γ r + γ ( 2 γ − 1 ) ∫ 1 ∞ u 2 γ − r − 2 d u = K 10 .$
Let further $1 ≤ γ ≤ 2$. By virtue of the inequalities
$( e − u γ + u γ − 1 ) u − 2 ≤ 1 , 1 − e − u γ u γ − 2 ≤ 1 ,$
$| ( γ − 1 ) u γ − 2 ( e − u γ − 1 ) − γ u 2 γ − 2 e − u γ | ≤ 2 γ − 1 ,$
we have
$∫ 1 ∞ | μ ″ ( u ) | d u ≤ r ( r + 1 ) 2 + 2 γ r + γ ( 2 γ − 1 ) ∫ 1 ∞ u − r d u = K 11 .$
Therefore, combining the relations (23), (24), we obtain
$∫ 1 ∞ μ ″ ( u ) d u = O ( 1 ) , δ → ∞ .$
In view of (16), taking into account (20), (22) and (25), we obtain
$∫ | t | ≥ δ γ | ∫ 0 ∞ μ ( u ) cos u t + β π 2 d u | d t = O ( 1 ) , γ < r < 2 γ , O ln δ , r = 2 γ , O δ r γ − 2 , r > 2 γ .$
Let us further consider
$∫ 0 δ γ | ∫ 0 ∞ μ ( u ) cos u t + β π 2 d u | d t ≤ ∫ 0 δ γ | ∫ 0 1 δ γ μ ( u ) cos u t + β π 2 d u | d t + + ∫ 0 δ γ | ∫ 1 δ γ ∞ μ ( u ) cos u t + β π 2 d u | d t .$
By the inequality (19), one can easily verify that the following relations hold
$∫ 0 δ γ | ∫ 0 1 δ γ μ ( u ) cos u t + β π 2 d u | d t ≤ ∫ 0 δ γ ∫ 0 1 δ γ μ ( u ) d u d t = K 12 δ r γ − 2 .$
The function $μ ( u )$ is monotonically decreasing on the interval $u 0 , ∞ , u 0 ≥ 1$, non-negative and tends to zero as $u → ∞$. Then, by the equality (4.16) from ([1], p. 69), we obtain
$∫ 0 δ γ | ∫ 1 δ γ ∞ μ ( u ) cos u t + β π 2 d u | d t = ∫ 0 δ γ | ∫ 1 δ γ ∞ μ ( u ) cos u t + β π 2 d u | d t ≤ ≤ ∫ 0 δ γ | ∫ 1 δ γ u 0 μ ( u ) cos u t + β π 2 d u | d t + ∫ 0 δ γ | ∫ u 0 ∞ μ ( u ) cos u t + β π 2 d u | d t ≤ ≤ ∫ 0 δ γ ∫ 1 δ γ u 0 μ ( u ) d u d t + ∫ 0 δ γ ∫ u 0 u 0 + 2 π t μ ( u ) d u d t ≤ ∫ 0 δ γ ∫ 1 δ γ u 0 + 2 π t μ ( u ) d u d t .$
Let $n ∈ N$ is such that $1 δ γ + 2 π n − 1 t ≤ u 0 ≤ 1 δ γ + 2 π n t$, then
$∫ 0 δ γ ∫ 1 δ γ u 0 + 2 π t μ ( u ) d u d t ≤ ∫ 0 δ γ ∫ 1 δ γ 1 δ γ + 2 π n + 1 t μ ( u ) d u d t .$
We transform the latter integral using a change of variable and integration by parts (assume that $δ > ( 2 π ( n + 1 ) + 1 ) γ$)
$∫ 0 δ γ ∫ 1 δ γ 1 δ γ + 2 π n + 1 t μ ( u ) d u d t = 2 π ( n + 1 ) ∫ 2 π n + 1 δ γ ∞ ∫ 1 δ γ 1 δ γ + x μ ( u ) d u d x x 2 = = 2 π n + 1 − 1 x ∫ 1 δ γ 1 δ γ + x μ ( u ) d u | 2 π ( n + 1 ) δ γ ∞ + ∫ 2 π n + 1 δ γ ∞ 1 x μ 1 δ γ + x d x = = 2 π ( n + 1 ) − lim x → ∞ 1 x ∫ 1 δ γ 1 δ γ + x e − u γ + u γ − 1 u − r d u + + δ γ 2 π n + 1 ∫ 1 δ γ 1 + 2 π n + 1 δ γ e − u γ + u γ − 1 u − r d u + + ∫ 2 π ( n + 1 ) δ γ 1 − 1 δ γ 1 x e − 1 δ γ + x γ + 1 δ γ + x γ − 1 1 δ γ + x − r d x + + ∫ 1 − 1 δ γ ∞ 1 x e − 1 δ γ + x γ + 1 δ γ + x γ − 1 1 δ γ + x − r d x .$
Obviously,
$lim x → ∞ 1 x ∫ 1 δ γ 1 δ γ + x e − u γ + u γ − 1 ψ ( δ γ u ) d u = 0 .$
Since the second inequality from (19) holds, then
$δ γ 2 π n + 1 ∫ 1 δ γ 1 + 2 π ( n + 1 ) δ γ e − u γ + u γ − 1 u − r d u ≤ ≤ δ γ 4 π n + 1 ∫ 1 δ γ 1 + 2 π ( n + 1 ) δ γ u 2 γ − r d u ≤ δ r + 1 γ 4 π n + 1 ∫ 1 δ γ 1 + 2 π ( n + 1 ) δ γ u 2 γ d u ≤ K 13 δ r γ − 2 .$
Using the second inequality from (19), we have
$∫ 2 π ( n + 1 ) δ γ 1 − 1 δ γ 1 x e − 1 δ γ + x γ + 1 δ γ + x γ − 1 1 δ γ + x − r d x ≤ ≤ ∫ 2 π ( n + 1 ) δ γ 1 − 1 δ γ 1 x 1 δ γ + x 2 γ − r d x = δ r γ − 2 ∫ 2 π ( n + 1 ) δ γ 1 − 1 δ γ 1 x 1 + δ γ x 2 γ − r d x = = δ r γ − 2 ∫ 1 + 2 π ( n + 1 ) δ γ y 2 γ − r y − 1 d y = δ r γ − 2 ∫ 1 + 2 π ( n + 1 ) δ γ y 2 γ − 1 ψ ( y ) 1 + 1 y − 1 d y ≤ ≤ 1 + 1 2 π ( n + 1 ) δ r γ − 2 ∫ 1 + 2 π ( n + 1 ) δ γ y 2 γ − 1 − r d y = K 14 + K 15 δ r γ − 2 , r ≠ 2 γ , K 16 ln δ , r = 2 γ .$
Considering the inequality
$e − u γ + u γ − 1 ≤ u γ ,$
we have
$∫ 1 − 1 δ γ ∞ 1 x e − 1 δ γ + x γ + 1 δ γ + x γ − 1 1 δ γ + x − r d x ≤ ≤ ∫ 1 − 1 δ γ ∞ 1 x 1 δ γ + x γ − r d x = δ r γ − 1 ∫ 1 − 1 δ γ ∞ 1 x 1 + δ γ x γ − r d x = = δ r γ − 1 ∫ δ γ ∞ y γ − r y − 1 d y = δ r γ − 1 ∫ δ γ ∞ y γ − 1 − r 1 + 1 y − 1 d y ≤ ≤ 1 + 1 δ γ − 1 δ r γ − 1 ∫ δ γ ∞ y γ − 1 − r d y ≤ K 17 δ r γ − 1 ∫ δ γ ∞ y γ − 1 − r d y = K 18 .$
From (27), taking into account (28) and (29)–(35), we can write the estimation
$∫ 0 δ γ | ∫ 0 ∞ μ ( u ) cos u t + β π 2 d u | d t = O ( 1 ) , γ < r < 2 γ , O ln δ , r = 2 γ , O δ r γ − 2 , r > 2 γ .$
Similarly, we can show that
$∫ − δ γ 0 | ∫ 0 ∞ μ ( u ) cos u t + β π 2 d u | d t = O ( 1 ) , γ < r < 2 γ , O ln δ , r = 2 γ , O δ r γ − 2 , r > 2 γ .$
Combining Formulas (26), (36) and (37), we obtain
$A μ = O ( 1 ) , γ < r < 2 γ , O ln δ , r = 2 γ , O δ r γ − 2 , r > 2 γ .$
Similarly to [27] we can show that the following equality holds
$f ( x ) − P γ ( δ , f , x ) = 1 ( δ γ ) r ∫ − ∞ ∞ f β ψ x + t δ γ τ ^ β ( t ) d t ,$
where
$τ ^ β ( t ) = τ ^ ( t , β ) = 1 π ∫ 0 ∞ τ ( u ) cos u t + β π 2 d u .$
Thence
$E W β , ∞ r ; P γ ( δ ) C = sup f ∈ W β , ∞ r 1 ( δ γ ) r ∫ − ∞ ∞ f β r x + t δ γ τ ^ β ( t ) d t C = = sup f ∈ W β , ∞ r 1 ( δ γ ) r ∫ − ∞ ∞ f β r x + t δ γ φ ^ β ( t ) + μ ^ β ( t ) d t C ≤ ≤ sup f ∈ W β , ∞ r 1 ( δ γ ) r ∫ − ∞ ∞ f β r x + t δ γ φ ^ β ( t ) d t C + 1 ( δ γ ) r ∫ − ∞ ∞ μ ^ β ( t ) d t .$
Therefore,
$E W β , ∞ r ; P γ ( δ ) C = sup f ∈ W β , ∞ r 1 ( δ γ ) r ∫ − ∞ ∞ f β r x + t δ γ φ ^ β ( t ) d t C + O 1 ( δ γ ) r A ( μ ) .$
Similarly to the work [28], we can show that the Fourier series of the function $f φ ( x ) = ∫ − ∞ ∞ f β r x + t δ γ φ ^ β ( t ) d t$ has the form:
$S f φ = ∑ k = 1 ∞ k γ ( δ γ ) γ − r a k cos k x + b k sin k x ,$
where $a k , b k$ are the Fourier coefficients of the function f. Therefore
$∫ − ∞ ∞ f β r x + t δ γ φ ^ β ( t ) d t = 1 ( δ γ ) γ − r f 0 γ ( x ) ,$
where $f 0 γ ( x )$ is $r , β$—derivative in the Weyl-Nagy sense for $r = γ , β = 0$.
Substituting (40) into (39), we obtain
$E W β , ∞ r ; P γ ( δ ) C = 1 δ sup f ∈ W β , ∞ r f 0 γ ( · ) C + O 1 ( δ γ ) r A ( μ ) , δ → ∞ .$
Substituting (38) into (41), we obtain the equation (5). The theorem is proved. □

## 3. Conclusions

One of the extremal problems of approximation theory, namely the problem of studying the asymptotic properties of linear summation methods of Fourier series, has been considered in the paper. Among the linear summation methods, on the one hand, there are methods that are defined by infinite numerical matrices, and on the other hand, methods that are defined by the set of functions of the natural argument that depend on the real parameter $δ$. This work is devoted to the study of the approximation properties of the methods of the last type, namely, generalized Poisson integrals. The Kolmogorov–Nikol’skii problem takes a special place among the extremal problems of the approximation theory. We have considered the problem of asymptotic equalities finding for the value of the exact upper limits of deviations of generalized Abel-Poisson integrals from functions of the Weyl-Nagy classes in the uniform metric. In particular, the asymptotic equality (5) for arbitrary $r > γ , 0 < γ ≤ 2$, has been written in the paper, providing the solution of the corresponding Kolmogorov-Nikol’skii problem. The importance of this type of problems in the theory of decision making, in signal transmission, in the study of mathematical models and in the coding and reproduction of images has been noted. Regarding further research in this direction, we note that similar problems can be considered in the broader classes of functions, such as Stepanets classes and classes of non-periodic locally summable functions.

## Author Contributions

Conceptualization, I.K. and Y.K.; methodology, I.K. and Y.K.; formal analysis, I.K. and Y.K.; writing—original draft preparation, I.K. and Y.K.; writing—review and editing, I.K. and Y.K. All authors have read and agreed to the published version of the manuscript.

## Funding

This work was supported by Grant of the Ministry of Education and Science of Ukraine, 0120U102630.

Not applicable.

Not applicable.

Not applicable.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Stepanets, A.I. Classification and Approximation of Periodic Functions; Kluwer: Dordrecht, The Netherlands, 1995. [Google Scholar]
2. Falaleev, L.P. On approximation of functions by generalized Abel-Poisson operators. Sib. Math. J. 2001, 42, 779–788. [Google Scholar] [CrossRef]
3. Bugrov, S. Inequalities of the type of Bernstein inequalities and their application to the investigation of the differential properties of the solutions of differential equations of higher order. Mathematica 1963, 5, 5–25. [Google Scholar]
4. Natanson, I.P. On the order of approximation of a continuous 2π-periodic function by its Poisson integral. Dokl. Akad. Nauk SSSR 1950, 72, 11–14. (In Russian) [Google Scholar]
5. Baskakov, V.A. Some properties of operators of Abel-Poisson type. Math. Notes 1975, 17, 101–107. [Google Scholar] [CrossRef]
6. Stepanets, A.I. Methods of Approximation Theory; VSP: Leiden, The Netherlands; Boston, MA, USA, 2005. [Google Scholar]
7. Shtark, E.L. Complete asymptotic expansion for the upper bound of the deviation of functions from Lip1 a singular Abel-Poisson integral. Math. Notes 1973, 13, 21–28. [Google Scholar]
8. Timan, A.F. Exact estimate of the remainder as approximation of periodic differentiable functions by Poisson integrals. Dokl. AN SSSR 1950, 74, 17–20. [Google Scholar]
9. Kharkevych, Y.I. On approximation of the quasi-smooth functions by their Poisson type integrals. J. Autom. Inf. Sci. 2017, 49, 74–81. [Google Scholar] [CrossRef]
10. Zhyhallo, T.V.; Padalko, N.I. On Some Boundary Properties of the Abel-Poisson Integrals. J. Autom. Inf. Sci. 2020, 52, 73–80. [Google Scholar] [CrossRef]
11. Kal’chuk, I.V.; Kharkevych, Y.I.; Pozharska, K.V. Asymptotics of approximation of functions by conjugate Poisson integrals. Carpathian Math. Publ. 2020, 12, 138–147. [Google Scholar] [CrossRef]
12. Korovkin, P.P. On the best approximation of functions of class Z2 by some linear operators. Dokl. Akad. Nauk SSSR 1959, 127, 143–149. (In Russian) [Google Scholar]
13. Bausov, L.I. Approximation of functions of class Zα by positive methods of summation of Fourier series. Uspekhi Mat. Nauk 1961, 16, 143–149. [Google Scholar]
14. Shvai, O.L.; Pozharska, K.V. On some approximation properties of Gauss-Weierstrass singular operators. J. Math. Sci. 2022, 260, 693–699. [Google Scholar] [CrossRef]
15. Zhyhallo, K.N. Complete asymptotics of approximations by certain singular integrals in mathematical modeling. J. Autom. Inf. Sci. 2020, 52, 58–68. [Google Scholar] [CrossRef]
16. Zhyhallo, T.V. Approximation in the mean of classes of the functions with fractional derivatives by their Abel-Poisson integrals. J. Autom. Inf. Sci. 2019, 51, 58–69. [Google Scholar] [CrossRef]
17. Hrabova, U.Z. Uniform approximations by the Poisson threeharmonic integrals on the Sobolev classes. J. Autom. Inf. Sci. 2019, 51, 46–55. [Google Scholar] [CrossRef]
18. Fang, X.; Deng, Y. Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement. Inverse Probl. Imaging 2018, 12, 733–743. [Google Scholar] [CrossRef] [Green Version]
19. Diao, H.; Cao, X.; Liu, H. On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications. Comm. Partial Differ. Equ. 2021, 46, 630–679. [Google Scholar] [CrossRef]
20. Sobchuk, V.; Kal’chuk, I.; Kharkevych, G.; Laptiev, O.; Kharkevych, Y.; Makarchuk, A. Solving the problem of convergence of the results of analog signals conversion in the process of aircraft control. In Proceedings of the 2021 IEEE 6th International Conference on Actual Problems of Unmanned Aerial Vehicles Development (APUAVD), Kyiv, Ukraine, 19–21 October 2021; pp. 29–32. [Google Scholar]
21. Cao, X.; Diao, H.; Li, J. Some recent progress on inverse scattering problems within general polyhedral geometry. Electron. Res. Arch. 2021, 29, 1753–1782. [Google Scholar] [CrossRef]
22. Hrabova, U.Z. Uniform approximations of functions of Lipschitz class by threeharmonic Poisson integrals. J. Autom. Inf. Sci. 2017, 49, 57–70. [Google Scholar] [CrossRef]
23. Serdyuk, A.S.; Stepanyuk, T.A. Uniform approximations by Fourier sums in classes of generalized Poisson integrals. Anal. Math. 2019, 45, 201–236. [Google Scholar] [CrossRef] [Green Version]
24. Pozhars’ka, K.V. Estimates for the entropy numbers of the classes $B p , θ Ω$ of periodic multivariable functions in the uniform metric. Ukr. Math. J. 2018, 70, 1439–1455. [Google Scholar] [CrossRef]
25. Chow, Y.T.; Deng, Y.; He, Y.; Liu, H.; Wang, X. Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes. SIAM J. Imaging Sci. 2021, 14, 946–975. [Google Scholar] [CrossRef]
26. Kal’chuk, I.V.; Kravets, V.I.; Hrabova, U.Z. Approximation of the classes $W β r H α$ by three-harmonic Poisson integrals. J. Math. Sci. 2020, 246, 39–50. [Google Scholar] [CrossRef]
27. Hrabova, U.Z.; Kal’chuk, I.V. Approximation of the classes $W β , ∞ r$ by three-harmonic Poisson integrals. Carpathian Math. Publ. 2019, 11, 321–334. [Google Scholar] [CrossRef] [Green Version]
28. Abdullayev, F.G.; Kharkevych, Y.I. Approximation of the classes $C β ψ H α$ by biharmonic Poisson integrals. Ukr. Math. J. 2020, 72, 21–38. [Google Scholar] [CrossRef]
 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Share and Cite

MDPI and ACS Style

Kal’chuk, I.; Kharkevych, Y. Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes. Axioms 2022, 11, 161. https://doi.org/10.3390/axioms11040161

AMA Style

Kal’chuk I, Kharkevych Y. Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes. Axioms. 2022; 11(4):161. https://doi.org/10.3390/axioms11040161

Chicago/Turabian Style

Kal’chuk, Inna, and Yurii Kharkevych. 2022. "Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes" Axioms 11, no. 4: 161. https://doi.org/10.3390/axioms11040161

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.