Approximation of Functions of the Classes ${\mathit{C}}_{\mathit{\beta }}^{\mathit{\psi }}{\mathit{H}}^{\mathit{\alpha }}$ by Linear Methods Summation of Their Fourier Series

Abstract

In this paper, we considered arbitrary linear summation methods of Fourier series specified by a set of continuous functions dependent on the real parameter and established their approximation properties. We obtained asymptotic formulas for the exact upper bounds of the deviations of operators generated by $\lambda$-methods of Fourier series summation from the functions of the classes $C_{\beta }^{\psi }{H}^{\alpha }$ under certain restrictions on the functions $\psi$.

1. Introduction

Let $f\left(x\right)$ be $2\pi$-periodic function from L, the Fourier series of which has the form

$$S\left[f\right]=\frac{a_0}{2}+\sum _{k=1}^{\infty }\left(a_{k}coskx+b_{k}sinkx\right),$$

where $a_k=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)coskx,k=0,1,\dots$, and $b_k=\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)sinkx,k=1,2,\dots$, are the Fourier coefficients of function f.

Let $\psi \left(k\right)$ be an arbitrary function of the natural argument and $\beta$ be an arbitrary fixed real number ($\beta \in R$). If the series

$$\sum _{k=1}^{\infty }\frac{1}{\psi \left(k\right)}\left(a_{k}cos\left(kx+\frac{\pi \beta }{2}\right)+b_{k}sin\left(kx+\frac{\pi \beta }{2}\right)\right)$$

is the Fourier series of some summable function $\phi$, then this function, according to O.I. Stepanets [1] (p. 25), is called the $(\psi ,\beta )$-derivative of the function f and is denoted by $f_{\beta }^{\psi }$. The set of all functions f, satisfying this condition, is denoted by $L_{\beta }^{\psi }$. If $f\in L_{\beta }^{\psi }$ and, in addition, $f_{\beta }^{\psi }\in \mathfrak{N}$, where $\mathfrak{N}$ is some subset of functions from L, then it is written $f\in L_{\beta }^{\psi }\mathfrak{N}$. Subsets of continuous functions from $L_{\beta }^{\psi }$ ($L_{\beta }^{\psi }\mathfrak{N}$) are denoted by $C_{\beta }^{\psi }$ ($C_{\beta }^{\psi }\mathfrak{N}$). When $\mathfrak{N}$ coincide with the set of functions $\phi$ that satisfy the condition $\mathrm{ess}sup\left|\phi \right|\le 1$, then the class $C_{\beta }^{\psi }\mathfrak{N}$ is denoted by $C_{\beta ,\infty }^{\psi }$.

We consider the set $C_{\beta }^{\psi }{H}^{\alpha }$ of functions f such that

$$C_{\beta }^{\psi }{H}^{\alpha }=\left\{f\left(x\right)\in C_{\beta }^{\psi },f_{\beta }^{\psi }\in H^{\alpha }:\left|f_{\beta }^{\psi }(x+h)-f_{\beta }^{\psi }\left(x\right)\right|\le {\left|h\right|}^{\alpha },h\in R,\right\},$$

where $0<\alpha \le 1$. We should note that when $\psi \left(k\right)=k^{-r},r>0$, then $C_{\beta }^{\psi }{H}^{\alpha }=W_{\beta }^r{H}^{\alpha }$, and $f_{\beta }^{\psi }\left(x\right)=f_{\beta }^r\left(x\right)$ is $(r,\beta )$-derivative in the Weyl–Nagy sense; if $r\in N$ and $r=\beta$, then $W_{\beta }^r{H}^{\alpha }=W^r{H}^{\alpha }$ is the class of functions $f\left(x\right)$ with a derivative of the order $r>0$ in Weyl sense; if $r\in N$ and $r=\beta +1$, then $W_{\beta }^r{H}^{\alpha }=\overline{W^r{H}^{\alpha }}$. Let us further define the class $C_{\beta }^{\psi }{H}^{\alpha }$ for $\alpha =0$, setting $C_{\beta }^{\psi }{H}^0=C_{\beta ,\infty }^{\psi }$.

Let $\Lambda =\left\{{\lambda }_{\delta }\left(u\right)\right\}$ be the set of continuous functions ${\lambda }_{\delta }\left(u\right)$ for all $u\ge 0$ such that ${\lambda }_{\delta }\left(0\right)=1$, and the parameter $\delta$ belongs to the set $E_{\Lambda }\subseteq \mathbb{R}\setminus \left\{0\right\}$ having the limit point ${\delta }_0$. Let us associate every function f using the set $\Lambda$ with the series

$$\frac{a_0}{2}{\lambda }_{\delta }\left(0\right)+\sum _{k=1}^{\infty }{\lambda }_{\delta }\left(\frac{k}{\delta }\right)\left(a_{k}coskx+b_{k}sinkx\right),$$

where $a_0,a_k,$ and $b_k$ are Fourier coefficients of the function f. If this series for all $\delta \in E_{\Lambda }$ is the Fourier series of some continuous function $U_{\delta }(f,x,\Lambda )$, then the set of functions $\Lambda$ is said to define the method of Fourier series summation of the function f.

The problem of finding asymptotic equalities for the quantities

$$\mathcal{E}{\left(\mathfrak{N},U_{\delta }(f;\Lambda )\right)}_X=\underset{f\in \mathfrak{N}}{sup}{∥f\left(x\right)-U_{\delta }(f,x,\Lambda )∥}_X,$$

(1)

where X is the normed space, $\mathfrak{N}\subseteq X$ is the given class of functions, and $U_{\delta }(f,x,\Lambda )$, $\delta \in E_{\Lambda }$ are polynomials determined by the specific method $U_{\delta }(f;\Lambda )$ of the series summation, is called (see, e.g., [2] (p. 198)) the Kolmogorov–Nikol’skii problem.

Let $\mathfrak{M}$ be the set of positive, continuous, convex downward functions $\psi \left(u\right),u\ge 1,$ that satisfy the condition

$$\underset{u\to \infty }{lim}\psi \left(u\right)=0.$$

Following O.I. Stepanets, [1] (p. 93) we associate each function, $\psi \in \mathfrak{M}$, with the pair of functions

$$\eta \left(t\right):={\psi }^{-1}\left(\psi \left(t\right)/2\right),\mu \left(t\right)=\mu (\psi ;t):=\frac{t}{\eta \left(t\right)-t},$$

where ${\psi }^{-1}(·)$ is the function inverse to $\psi (·)$. Then, according to [2] (pp. 159–160), we set

$${\mathfrak{M}}_0=\left\{\psi \in \mathfrak{M}:0<\mu (\psi ;t)\le K\forall t\ge 1\right\}$$

and

$${\mathfrak{M}}_C=\left\{\psi \in \mathfrak{M}:0<K_1\le \mu (\psi ;t)\le K_2\forall t\ge 1\right\},$$

where $K,K_i$, $i=1,2$ are constants that may depend on $\psi$.

We consider some function $\tau \left(u\right)$ continuous on $[0;\infty )$, given in the form

$$\tau \left(u\right)={\tau }_{\delta }\left(u\right)=\left\{\begin{array}{cc}\left(1-{\lambda }_{\delta }\left(u\right)\right)\frac{\psi \left(1\right)}{\psi \left(\delta \right)},\hfill & 0\le u\le \frac{1}{\delta },\hfill \\ \left(1-{\lambda }_{\delta }\left(u\right)\right)\frac{\psi \left(\delta u\right)}{\psi \left(\delta \right)},\hfill & u\ge \frac{1}{\delta },\hfill \end{array}\right.$$

(2)

the Fourier transform of which,

$$\widehat{\tau }\left(t\right)=\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos(vt+\frac{\beta \pi }{2})du,$$

(3)

is summable on the whole real axis.

In this work, we study the quantities

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C=\underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥f\left(x\right)-U_{\delta }(f,x,\Lambda )∥}_C,$$

where $U_{\delta }(f,\Lambda )$ are operators defined by the set $\Lambda$, the limit point of which, ${\delta }_0$, is equal to ∞ (${\delta }_0=\infty$), and ${∥f∥}_C=\underset{x\in R}{max}\left|f\left(x\right)\right|$.

Asymptotic equalities for quantities of type (1) for various linear methods of summation of Fourier series on classes of differentiable functions of one and many variables have been obtained in [2,3,4,5,6]. We should note that in the vast majority of works, the triangular case of numerical matrices $\Lambda$ has been considered. The results of L.I. Bausov’s work [7,8] for classes $W_{\beta }^r$ and $W_{\beta }^r{H}^{\alpha }$ are adapted to the case of rectangular numerical matrices. The approximate properties of linear summation methods that are determined by the set of functions $\Lambda =\left\{{\lambda }_{\delta }\left(u\right)\right\}$ continuous on $[0,\infty )$ and dependent on the real parameter $\delta$, are well studied on the classes $W_{\beta ,\infty }^r$, $W^r{H}_{\omega }$, and $L_{\beta ,1}^{\psi }$ in [9,10,11,12,13,14], and at the same time, a similar problem for the classes $C_{\beta }^{\psi }{H}^{\alpha }$ remains open.

The Fourier series summation methods have been actively developing recently and are widely used in various fields, including signal processing [15,16], computer graphics, cyberphysical systems [17], machine learning [18], mathematical modelling [19,20], etc. We should note that linear summation methods of Fourier series have a number of advantages over traditional Fourier series summation methods: first, they can be more accurate than traditional methods; secondly, they can be more resistant to noise; and third, they can be more adaptive to different signals. This is extremely important when modeling processes for which the functional stability of complex technical systems must be guaranteed [21,22,23]. Methods of constructing the asymptotic solution of differential equations are important for hydrodynamic problems [24], for mathematical models of processes that describe phenomena where the action of internal and external destabilizing factors of an impulsive nature takes place [25,26], for heat conduction equations [27], etc.

2. Main Result

Let us give the definitions we need.

Definition 1

(L.I. Bausov [8]). Let the function $\tau \left(u\right)$ be given on $[0,\infty )$, being absolutely continuous, $\tau (\infty )=0$ and $0\le \alpha <1$. The function $\tau \left(u\right)\in {\mathcal{E}}_{\alpha }$ if the derivative, ${\tau }^{\prime }\left(u\right)$, at those points where it does not exist can be defined so that for some $a\ge 0$ the following integrals

$$\underset{0}{\overset{\frac{a}{2}}{\int }}{u}^{1-\alpha }|d{\tau }^{\prime }\left(u\right)|,\underset{\frac{a}{2}}{\overset{\frac{3a}{2}}{\int }}{|u-a|}^{1-\alpha }|d{\tau }^{\prime }\left(u\right)|,\underset{\frac{3a}{2}}{\overset{\infty }{\int }}(u-a)\left|d{\tau }^{\prime }\left(u\right)\right|.$$

are convergent.

Definition 2.

Let the function ${\tau }_{\delta }\left(u\right)$ be defined by equality (2). Let us say that ${\tau }_{\delta }\left(u\right)\in B_{\alpha }$ if the following conditions are satisfied:

(1) ${\tau }_{\delta }\left(u\right)\in {\mathcal{E}}_{\alpha }$, $0\le \alpha <1$;

(2) The Fourier transform $\widehat{\tau }\left(t\right)$ of form (3) is summable and preserves the sign on each of the intervals, $(-\theta ,0)$ and $(0,\theta )$, where $\theta ={\theta }_{\delta }$ is some number, $0<\theta \le \frac{\pi }{2}$;

(3) the integral

$$A(\alpha ,\tau )=\frac{1}{\pi }\underset{-\infty }{\overset{\infty }{\int }}{\left|t\right|}^{\alpha }\left|\underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)du\right|dt$$

(4)

is convergent.

The following theorem holds.

Theorem 1.

Let ${\tau }_{\delta }\left(u\right)\in B_{\alpha }$, $0\le \alpha <1$; then, at $\delta \to \infty$

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C=\frac{\gamma }{{\delta }^{\alpha }}\psi \left(\delta \right)A(\alpha ,{\tau }_{\delta })+O\left(\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}a(\alpha ,{\tau }_{\delta })\right),$$

(5)

where

$$a(\alpha ,{\tau }_{\delta })=\underset{\left|t\right|\ge \delta \theta }{\int }{\left|t\right|}^{\alpha }\left|\underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)du\right|dt,$$

(6)

$$2^{\alpha -1}\le \gamma =\gamma \left(\alpha \right)\le 1.$$

Proof. 

Let $\delta \in E_{\Lambda }$, $x\in \mathbb{R}$. Let us define

$$T_{\beta ,\delta }^{\psi }\left(x\right)=\underset{-\infty }{\overset{\infty }{\int }}{f}_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)\widehat{\tau }\left(t\right)dt.$$

Repeating the considerations given by O.I. Stepanets [1] (p. 54) when finding integral formulas for the quantities $f\left(x\right)-U_n(f;x;\Lambda )$, it is easy to show that the $2\pi$-periodic function $T_{\beta ,\delta }^{\psi }\left(x\right)$ is continuous and the following equality holds for it:

$$S\left[T_{\beta ,\delta }^{\psi }\right]=\sum _{k=0}^{\infty }\tau \left(\frac{k}{\delta }\right)\frac{1}{\psi \left(k\right)}(a_{k}coskx+b_{k}sinkx).$$

Hence, taking into account (2), we have

$$S\left[\underset{-\infty }{\overset{\infty }{\int }}{f}_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)\widehat{\tau }\left(t\right)dt\right]=\sum _{k=0}^{\infty }\frac{1-{\lambda }_{\delta }\left(\frac{k}{\delta }\right)}{\psi \left(\delta \right)}(a_{k}coskx+b_{k}sinkx)$$

$$=S\left[\frac{1}{\psi \left(\delta \right)}\left(f\left(x\right)-U_{\delta }(f,x,\Lambda )\right)\right].$$

Thus, for any $f\in C_{\beta }^{\psi }{H}^{\alpha }$ and for the function $\tau \left(u\right)$ of form (2), due to the equality

$$cos\frac{\beta \pi }{2}\underset{-\infty }{\overset{\infty }{\int }}\widehat{\tau }\left(t\right)dt=\tau \left(0\right)$$

(see [1] (p. 52)), we have

$$f\left(x\right)-U_{\delta }(f;x;\Lambda )=\frac{\psi \left(\delta \right)}{\pi }\underset{-\infty }{\overset{\infty }{\int }}\left[f_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)-f_{\beta }^{\psi }\left(x\right)\right]$$

$$\times \underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)dvdt$$

(7)

Since $f_{\beta }^{\psi }\in H^{\alpha }$, from relation (7), it immediately follows that

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C\le \frac{\psi \left(\delta \right)}{\pi {\delta }^{\alpha }}\underset{-\infty }{\overset{\infty }{\int }}{\left|t\right|}^{\alpha }\left|\underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)du\right|dt.$$

(8)

Let us consider the function

$$g\left(x\right)=\left\{\begin{array}{cc}{\chi \left|x\right|}^{\alpha }\mathrm{sign}\widehat{\tau }\left(\delta x\right),\hfill & \left|x\right|<\theta ,\hfill \\ \hfill \\ \chi {(\pi -x)}^{\alpha }{\left(\frac{\theta }{\pi -\theta }\right)}^{\alpha }\mathrm{sign}\widehat{\tau }\left(\frac{\theta }{2}\right),\hfill & x\in [\theta ,\pi ],\hfill \\ \hfill \\ \chi {(\pi +x)}^{\alpha }{\left(\frac{\theta }{\pi -\theta }\right)}^{\alpha }\mathrm{sign}\widehat{\tau }\left(-\frac{\theta }{2}\right),\hfill & x\in [-\pi ,-\theta ],\hfill \end{array}\right.$$

(9)

where $\chi =1$ at $\widehat{\tau }\left(\frac{\theta }{2}\right)\widehat{\tau }\left(-\frac{\theta }{2}\right)>0$, $\chi =2^{\alpha -1}$ at $\widehat{\tau }\left(\frac{\theta }{2}\right)\widehat{\tau }\left(-\frac{\theta }{2}\right)<0,$ $g(x+2\pi )=g\left(x\right)$, and the function g, as shown in [8], belongs to the class $H^{\alpha }$. Let us put

$$F\left(x\right)=\frac{a_0\left(F\right)}{2}+\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}g(x+t)\sum _{k=1}^{\infty }\psi \left(k\right)cos\left(kt+\frac{\beta \pi }{2}\right)dt.$$

The function $F\left(x\right)\in C_{\beta }^{\psi }{H}^{\alpha }$, since, as it follows from [2], $F_{\beta }^{\psi }(·)=g(·)$. Then, from (7) and (9) we obtain

$$F\left(0\right)-U_{\delta }(F;0;\Lambda )=\frac{\psi \left(\delta \right)}{\pi }\underset{-\infty }{\overset{\infty }{\int }}g\left(\frac{t}{\delta }\right)\underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)dvdt$$

$$=\psi \left(\delta \right)\left(\underset{-\delta \theta }{\overset{\delta \theta }{\int }}g\left(\frac{t}{\delta }\right)\widehat{\tau }\left(t\right)dt+\underset{\left|t\right|\ge \delta \theta }{\int }g\left(\frac{t}{\delta }\right)\widehat{\tau }\left(t\right)dt\right)$$

$$=\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}\underset{-\delta \theta }{\overset{\delta \theta }{\int }}\chi {\left|t\right|}^{\alpha }\mathrm{sign}\widehat{\tau }\left(t\right)\widehat{\tau }\left(t\right)dt+\psi \left(\delta \right)\underset{\left|t\right|\ge \delta \theta }{\int }g\left(\frac{t}{\delta }\right)\widehat{\tau }\left(t\right)dt.$$

From here and from (6), taking into account that ${\tau }_{\delta }\left(u\right)\in B_{\alpha }$, we obtain

$$\left|F\left(0\right)-U_{\delta }(F;0;\Lambda )\right|\le \frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}\chi \underset{-\infty }{\overset{\infty }{\int }}{\left|t\right|}^{\alpha }|\widehat{\tau }\left(t\right)|dt+\psi \left(\delta \right)\underset{\left|t\right|\ge \delta \theta }{\int }\left|g\left(\frac{t}{\delta }\right)\right|\left|\widehat{\tau }\left(t\right)\right|dt$$

$$=\frac{\chi \psi \left(\delta \right)}{\pi {\delta }^{\alpha }}\underset{-\infty }{\overset{\infty }{\int }}{\left|t\right|}^{\alpha }\left|\underset{0}{\overset{\infty }{\int }}\tau \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)du\right|dt+O\left(\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}a(\alpha ,{\tau }_{\delta })\right).$$

(10)

From relations (8) and (10), due to the invariance of the classes $C_{\beta }^{\psi }{H}^{\alpha }$ with respect to the shift of the argument [1] (p. 109), we obtain the statement of the theorem. Theorem 1 is proved. □

Let

$$f_{\phi }\left(x\right)=\underset{-\infty }{\overset{\infty }{\int }}\left(f_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)-f_{\beta }^{\psi }\left(x\right)\right)\widehat{\phi }\left(t\right)dt,$$

where $\widehat{\phi }\left(t\right)$ is the Fourier transform of the function $\phi$ of the form

$$\widehat{\phi }\left(t\right)=\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\phi \left(u\right)cos\left(ut+\frac{\beta \pi }{2}\right)du.$$

Theorem 2.

Let ${\tau }_{\delta }\left(u\right)={\phi }_{\delta }\left(u\right)+{\mu }_{\delta }\left(u\right)$ and the functions $\phi \left(u\right)={\phi }_{\delta }\left(u\right)$ and $\mu \left(u\right)={\mu }_{\delta }\left(u\right)$ satisfy the condition $B_{\alpha }$, $0\le \alpha <1$; then, the equality

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C=\psi \left(\delta \right)\underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥f_{\phi }\left(x\right)∥}_C+O\left(\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}A(\alpha ,{\mu }_{\delta })\right)$$

(11)

holds at $\delta \to \infty$, where $A(\alpha ,{\mu }_{\delta })$ is defined by equality (4).

Proof. 

From equality (7), taking into account that $f_{\beta }^{\psi }\in H^{\alpha }$, we have

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;x;\Lambda )\right)}_C=\underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥f\left(x\right)-U_{\delta }(f,x,\Lambda )∥}_C$$

$$=\underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥\psi \left(\delta \right)\underset{-\infty }{\overset{\infty }{\int }}\left(f_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)-f_{\beta }^{\psi }\left(x\right)\right)\widehat{\tau }\left(t\right)dt∥}_C$$

$$=\underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥\psi \left(\delta \right)\underset{-\infty }{\overset{\infty }{\int }}\left(f_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)-f_{\beta }^{\psi }\left(x\right)\right)(\widehat{\phi }\left(t\right)+\widehat{\mu }\left(t\right))dt∥}_C$$

$$\le \underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥\psi \left(\delta \right)\underset{-\infty }{\overset{\infty }{\int }}\left(f_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)-f_{\beta }^{\psi }\left(x\right)\right)\widehat{\phi }\left(t\right)dt∥}_C+\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}\underset{-\infty }{\overset{\infty }{\int }}{\left|t\right|}^{\alpha }\left|\widehat{\mu }\left(t\right)\right|dt.$$

From here and from (4), we obtain

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C=\underset{f\in C_{\beta }^{\psi }{H}^{\alpha }}{sup}{∥\psi \left(\delta \right)\underset{-\infty }{\overset{\infty }{\int }}\left(f_{\beta }^{\psi }\left(x+\frac{t}{\delta }\right)-f_{\beta }^{\psi }\left(x\right)\right)\widehat{\phi }\left(t\right)dt∥}_C$$

$$+O\left(\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}A(\alpha ,{\mu }_{\delta })\right),\delta \to \infty .$$

Thus, from the last formula for the $2\pi$-periodic continuous function $f_{\phi }$, relation (11) holds. Theorem 2 is proved. □

Theorem 3.

Let ${\tau }_{\delta }\left(u\right)\in B_{\alpha }$, $0\le \alpha <1$, $\psi \left(u\right)\in {\mathfrak{M}}_0$, and

$$\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|\le \left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|,u\in \left[0;a\right].$$

(12)

Then, as $\delta \to \infty$,

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C=\frac{2\gamma \left|sin\frac{\beta \pi }{2}\right|}{\pi {\delta }^{\alpha }}\psi \left(\delta \right)\underset{0}{\overset{\infty }{\int }}\left|{\tau }_{\delta }\left(u\right)\right|\frac{du}{u^{1+\alpha }}$$

$$+O\left(\frac{\psi \left(\delta a\right)}{{\delta }^{\alpha }}\underset{0}{\overset{a}{\int }}\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|\frac{du}{u^{1+\alpha }}\right)$$

$$+\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}O\left(\underset{0}{\overset{1/\delta \theta }{\int }}\left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|\frac{du}{u^{1+\alpha }}+H(\alpha ,{\tau }_{\delta })\right),$$

(13)

where $2^{\alpha -1}\le \gamma =\gamma \left(\alpha \right)\le 1$ and $H(\alpha ,\tau )$ is defined by the relation

$$H(\alpha ,\tau )=\frac{\left|\tau \left(0\right)\right|+\left|\tau \left(a\right)\right|}{a^{\alpha }}+\underset{0}{\overset{\frac{a}{2}}{\int }}{u}^{1-\alpha }\left|d{\tau }^{\prime }\left(u\right)\right|+\underset{\frac{a}{2}}{\overset{\frac{3a}{2}}{\int }}{\left|u-a\right|}^{1-\alpha }\left|d{\tau }^{\prime }\left(u\right)\right|$$

$$+\frac{1}{a^{\alpha }}\underset{\frac{3a}{2}}{\overset{\infty }{\int }}\left(u-a\right)\left|d{\tau }^{\prime }\left(u\right)\right|.$$

(14)

Proof. 

Since ${\tau }_{\delta }\left(u\right)\in B_{\alpha }$, according to Theorem 1, condition (5) holds, and due to Theorem 12.1 of [2] (p. 161), relation (1.11) of [8] is valid:

$$\left|A(\alpha ,\tau )-\frac{4}{{\pi }^2}\underset{0}{\overset{\infty }{\int }}\xi \left(sin\frac{\beta \pi }{2}\tau \left(u\right),j_u\left[\tau (a-u)-\tau (a+u)\right]\right)\frac{du}{u^{1+\alpha }}\right|\le KH(\alpha ,\tau ),$$

(15)

where $\xi (A,B)$ is the function introduced in [28], so we have

$$\xi (A,B)=\left\{\begin{array}{cc}\frac{\pi }{2}\left|A\right|,\hfill & \left|B\right|\le \left|A\right|,\hfill \\ \left|A\right|arcsin\left|\frac{A}{B}\right|+\sqrt{B^2-A^2},\hfill & \left|B\right|>\left|A\right|,\hfill \end{array}\right.$$

(16)

$$j_u=\left\{\begin{array}{cc}1,\hfill & 0<u<a,\hfill \\ 0,\hfill & u\ge a.\hfill \end{array}\right.$$

(17)

To estimate the integral $a(\alpha ,{\tau }_{\delta })$ from (5), we use equality (2.5) of [28] (p. 257):

$$4\xi (A,B)=\underset{-\pi }{\overset{\pi }{\int }}|A+Bsint|dt=\underset{-\pi }{\overset{\pi }{\int }}|A+Bcost|dt.$$

Then,

$$4\xi (sin\frac{\beta \pi }{2}\tau \left(u\right),j_u\left[\tau (a-u)-\tau (a+u)\right])\le 2\pi \left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|$$

$$+4j_u\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|.$$

Let us put into the conditions of Lemma 3 in [8] $\epsilon =\delta \theta$ at $\theta ={\theta }_{\delta }$ and $\delta \theta \to 0$. Then, using the last inequality, we obtain

$$a(\alpha ,{\tau }_{\delta })=\frac{1}{\pi }\underset{0}{\overset{1/\delta \theta }{\int }}\left(2\pi \left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|+4j_u\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|\right)\frac{du}{u^{1+\alpha }}$$

$$+O\left(H(\alpha ,{\tau }_{\delta })\right),\delta \to \infty .$$

(18)

Combining Formulas (5), (15), and (18), we obtain

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C$$

$$=\frac{4\gamma }{{\pi }^2{\delta }^{\alpha }}\psi \left(\delta \right)\underset{0}{\overset{\infty }{\int }}\xi \left(sin\frac{\beta \pi }{2}{\tau }_{\delta }\left(u\right),j_u\left[{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right]\right)\frac{du}{u^{1+\alpha }}+\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}$$

$$\times O\left(\underset{0}{\overset{1/\delta \theta }{\int }}\left(\left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|+j_u\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|\right)\frac{du}{u^{1+\alpha }}+H(\alpha ,{\tau }_{\delta })\right).$$

(19)

We show that when $\forall a>\frac{1}{2\delta }$, the formula

$$\underset{0}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du$$

$$=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{0}{\overset{a}{\int }}\frac{\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|}{u^{1+\alpha }}du+H(\alpha ,{\tau }_{\delta })O\left(1\right)$$

(20)

holds, where $O\left(1\right)$ is a value uniformly bounded by $\delta$.

From relation (2), we find

$${\tau }_{\delta }(a-u)=\left\{\begin{array}{cc}\left(1-{\lambda }_{\delta }(a-u)\right)\frac{\psi \left(1\right)}{\psi \left(\delta \right)},\hfill & a-\frac{1}{\delta }\le u\le a,\hfill \\ \left(1-{\lambda }_{\delta }(a-u)\right)\frac{\psi \left(\delta \right(a-u\left)\right)}{\psi \left(\delta \right)},\hfill & u\le a-\frac{1}{\delta },\hfill \end{array}\right.$$

(21)

$${\tau }_{\delta }(a+u)=\left\{\begin{array}{cc}\left(1-{\lambda }_{\delta }(a+u)\right)\frac{\psi \left(1\right)}{\psi \left(\delta \right)},\hfill & -a\le u\le \frac{1}{\delta }-a,\hfill \\ \left(1-{\lambda }_{\delta }(a+u)\right)\frac{\psi \left(\delta \right(a+u\left)\right)}{\psi \left(\delta \right)},\hfill & u\ge \frac{1}{\delta }-a.\hfill \end{array}\right.$$

(22)

Let $a>\frac{1}{\delta }$. The integral from the left side (20) is presented as the sum of two integrals:

$$\underset{0}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du=\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du$$

$$+\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du.$$

(23)

Let us first estimate the first term of the right-hand side (23). For this purpose, we add and subtract the value under the sign of the module in the integrand function

$$\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\left({\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right).$$

We obtain

$$\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|}{u^{1+\alpha }}du$$

$$+O\left(\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)+\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\left({\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right)\right|}{u^{1+\alpha }}du\right).$$

(24)

Since relations (21) and (22) hold, for $u\in \left[0,a-\frac{1}{\delta }\right]$,

$${\lambda }_{\delta }(a-u)=1-\frac{\psi \left(\delta \right)}{\psi \left(\delta \right(a-u\left)\right)}{\tau }_{\delta }(a-u)$$

and

$${\lambda }_{\delta }(a+u)=1-\frac{\psi \left(\delta \right)}{\psi \left(\delta \right(a+u\left)\right)}{\tau }_{\delta }(a+u).$$

Then,

$$\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)+\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\left({\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right)\right|}{u^{1+\alpha }}du$$

$$\le \underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\left|{\tau }_{\delta }(a-u)\right|\left|1-\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right(a-u\left)\right)}\right|\frac{du}{u^{1+\alpha }}$$

$$+\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\left|{\tau }_{\delta }(a+u)\right|\left|1-\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right(a+u\left)\right)}\right|\frac{du}{u^{1+\alpha }}.$$

(25)

Since ${\tau }_{\delta }\left(u\right)\in {\mathcal{E}}_{\alpha }$, according to Lemma 1 of [8], we obtain that

$$\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\left|{\tau }_{\delta }(a-u)\right|\left|1-\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right(a-u\left)\right)}\right|\frac{du}{u^{1+\alpha }}$$

$$+\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\left|{\tau }_{\delta }(a+u)\right|\left|1-\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right(a+u\left)\right)}\right|\frac{du}{u^{1+\alpha }}$$

$$=H(\alpha ,{\tau }_{\delta })O\left(\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|\psi \left(\delta \right(a-u\left)\right)-\psi \left(\delta a\right)\right|}{u^{1+\alpha }\psi \left(\delta (a-u)\right)}du+\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|\psi \left(\delta \right(a+u\left)\right)-\psi \left(\delta a\right)\right|}{u^{1+\alpha }\psi \left(\delta (a+u)\right)}du\right).$$

(26)

We show that as $\delta \to \infty$,

$$I_{1,\delta }:=\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|\psi \left(\delta \right(a-u\left)\right)-\psi \left(\delta a\right)\right|}{u^{1+\alpha }\psi \left(\delta (a-u)\right)}du=O\left(1\right)$$

(27)

and

$$I_{2,\delta }:=\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|\psi \left(\delta \right(a+u\left)\right)-\psi \left(\delta a\right)\right|}{u^{1+\alpha }\psi \left(\delta (a+u)\right)}du=O\left(1\right),$$

(28)

where $O\left(1\right)$ is a value uniformly bounded by $\delta$. Really, the function $\frac{\psi \left(\delta \right(a-u\left)\right)-\psi \left(\delta a\right)}{u\psi \left(\delta \right(a-u\left)\right)}$ is bounded for all $u\in \left[\epsilon ,a-\frac{1}{\delta }\right]$, $0<\epsilon <a-\frac{1}{\delta }$ and, moreover, taking into account Theorem 12.1 of [2] (p. 161),

$$\underset{u\to 0}{lim}\frac{1-\psi \left(\delta a\right)/\psi \left(\delta \right(a-u\left)\right)}{u}=\frac{\delta \left|{\psi }^{\prime }\left(\delta a\right)\right|}{\psi \left(\delta a\right)}\le K.$$

Then,

$$I_{1,\delta }\le \underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{K_1}{u^{\alpha }}du=\frac{K_1}{1-\alpha }{\left(a-\frac{1}{\delta }\right)}^{1-\alpha }\le K_2,\delta \to \infty ,a>\frac{1}{\delta }.$$

Therefore, $I_{1,\delta }=O\left(1\right)$ at $\delta \to \infty$.

Taking into account that $\psi \left(\delta \right(a+u\left)\right)<\psi \left(\delta a\right)$, similarly, we can shown that

$$I_{2,\delta }\le K\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{du}{u^{\alpha }}\le K_3.$$

Therefore, equalities (27) and (28) hold. Combining relations (24)–(28), we obtain

$$\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{0}{\overset{a-\frac{1}{\delta }}{\int }}\frac{\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|}{u^{1+\alpha }}du$$

$$+H(\alpha ,{\tau }_{\delta })O\left(1\right).$$

(29)

Let us estimate the second term from the right-hand side of equality (23). It is obvious that

$$\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|}{u^{1+\alpha }}du$$

$$+O\left(\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)+\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\left({\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right)\right|}{u^{1+\alpha }}du\right).$$

(30)

From relations (21) and (22), we have that for $u\in \left[a-\frac{1}{\delta },a\right]$:

$${\lambda }_{\delta }(a-u)=1-\frac{\psi \left(\delta \right)}{\psi \left(1\right)}{\tau }_{\delta }(a-u)$$

(31)

and

$${\lambda }_{\delta }(a+u)=1-\frac{\psi \left(\delta \right)}{\psi \left(\delta \right(a+u\left)\right)}{\tau }_{\delta }(a+u).$$

Hence, due to Lemma 1 of [8], we obtain that

$$\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)+\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\left({\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right)\right|}{u^{1+\alpha }}du$$

$$=\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)\left(1-\frac{\psi \left(\delta a\right)}{\psi \left(1\right)}\right)-{\tau }_{\delta }(a+u)\left(1-\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right(a+u\left)\right)}\right)\right|}{u^{1+\alpha }}du$$

$$=H(\alpha ,{\tau }_{\delta })O\left(\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|\psi \left(1\right)-\psi \left(\delta a\right)\right|}{u^{1+\alpha }\psi \left(1\right)}du+\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|\psi \left(\delta \right(a+u\left)\right)-\psi \left(\delta a\right)\right|}{u^{1+\alpha }\psi \left(\delta (a+u)\right)}du\right).$$

(32)

Let us estimate the right-hand side of equality (32). The functions

$$\frac{\psi \left(1\right)-\psi \left(\delta a\right)}{u^{1+\alpha }\psi \left(1\right)},\frac{\psi \left(\delta a\right)-\psi \left(\delta \right(a+u\left)\right)}{u^{1+\alpha }\psi \left(\delta (a+u)\right)}$$

for all $u\in \left[a-\frac{1}{\delta };a\right]$, $a>\frac{1}{\delta }$, are continuous and therefore bounded. Then,

$$\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\psi \left(1\right)-\psi \left(\delta a\right)}{u^{1+\alpha }\psi \left(1\right)}du+\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\psi \left(\delta a\right)-\psi \left(\delta \right(a+u\left)\right)}{u^{1+\alpha }\psi \left(\delta (a+u)\right)}du=O\left(1\right).$$

(33)

Due to relations (30)–(33), we obtain

$$\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}{u^{1+\alpha }}du=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{a-\frac{1}{\delta }}{\overset{a}{\int }}\frac{\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|}{u^{1+\alpha }}du$$

$$+H(\alpha ,{\tau }_{\delta })O\left(1\right).$$

(34)

Combining relations (34) and (29), we obtain equality (20). Similarly, as when proving relation (20) for $a>\frac{1}{\delta }$, we can show that equality (20) also holds for the case $\frac{1}{2\delta }<a\le \frac{1}{\delta }$.

Equality (13) is obtained on the basis of relation (19) using condition (12), the definition of the function $\xi (A,B)$, and formula (20). Theorem 3 is proved. □

Theorem 4.

Let ${\tau }_{\delta }\left(u\right)\in B_{\alpha }$, $0\le \alpha <1$, $\psi \left(u\right)\in {\mathfrak{M}}_0$, and

$$\left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|<\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|,u\in \left[0;a\right].$$

(35)

Then, for $\delta \to \infty$,

$$\mathcal{E}{\left(C_{\beta }^{\psi }{H}^{\alpha },U_{\delta }(f;\Lambda )\right)}_C=\frac{4\gamma }{{\pi }^2{\delta }^{\alpha }}\psi \left(\delta a\right)\underset{0}{\overset{a}{\int }}\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|\frac{du}{u^{1+\alpha }}+$$

$$+O\left(\frac{\psi \left(\delta a\right)}{{\delta }^{\alpha }}\underset{0}{\overset{1/\delta \theta }{\int }}\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|\frac{du}{u^{1+\alpha }}\right)$$

$$+\frac{\psi \left(\delta \right)}{{\delta }^{\alpha }}O\left(\left|sin\frac{\beta \pi }{2}\right|\underset{0}{\overset{\infty }{\int }}\left|{\tau }_{\delta }\left(u\right)\right|\frac{du}{u^{1+\alpha }}+H(\alpha ,{\tau }_{\delta })\right).$$

(36)

Proof. 

From relation (16), taking into account formula (35), we obtain

$$\underset{0}{\overset{\infty }{\int }}\xi \left(sin\frac{\beta \pi }{2}{\tau }_{\delta }\left(u\right),j_u\left[{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right]\right)\frac{du}{u^{1+\alpha }}$$

$$=\underset{0}{\overset{a}{\int }}\left(\left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|arcsin\frac{\left|sin\frac{\beta \pi }{2}\right|\left|{\tau }_{\delta }\left(u\right)\right|}{\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|}\right.$$

$$+\left.\sqrt{{\left({\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right)}^2-{sin}^2\frac{\beta \pi }{2}{\tau }_{\delta }^2\left(u\right)}\right)\frac{du}{u^{1+\alpha }}$$

$$+\underset{a}{\overset{\infty }{\int }}\xi \left(sin\frac{\beta \pi }{2}{\tau }_{\delta }\left(u\right),0\right)\frac{du}{u^{1+\alpha }}$$

$$=\underset{0}{\overset{a}{\int }}\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|{\left(1-\frac{{sin}^2\frac{\beta \pi }{2}{\tau }_{\delta }^2\left(u\right)}{{\left({\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right)}^2}\right)}^{1/2}\frac{du}{u^{1+\alpha }}$$

$$+O\left(\left|sin\frac{\beta \pi }{2}\right|\underset{0}{\overset{\infty }{\int }}\left|{\tau }_{\delta }\left(u\right)\right|\frac{du}{u^{1+\alpha }}\right),\delta \to \infty .$$

(37)

In the case when $u\in (0,a]$, we obtain from estimate (35) that

$$0<\frac{{sin}^2\frac{\beta \pi }{2}{\tau }_{\delta }^2\left(u\right)}{{\left({\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right)}^2}<1.$$

Then, from (37), using the power series expansion of the functions ${\left(1-u\right)}^{\frac{1}{2}}$, $u\in \left(0,1\right)$ ad taking into account (20), we have that

$$\underset{0}{\overset{\infty }{\int }}\xi \left(sin\frac{\beta \pi }{2}{\tau }_{\delta }\left(u\right),j_u\left[{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right]\right)\frac{du}{u^{1+\alpha }}$$

$$=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{0}{\overset{a}{\int }}\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|\frac{du}{u^{1+\alpha }}+O\left(\left|sin\frac{\beta \pi }{2}\right|\underset{0}{\overset{\infty }{\int }}\left|{\tau }_{\delta }\left(u\right)\right|\frac{du}{u^{1+\alpha }}\right)$$

$$+O\left(H(\alpha ,{\tau }_{\delta })\right).$$

Substituting the last estimate into (19) and relying on the fact that

$$\underset{0}{\overset{1/\delta \theta }{\int }}{j}_u\left|{\tau }_{\delta }(a-u)-{\tau }_{\delta }(a+u)\right|\frac{du}{u^{1+\alpha }}=\frac{\psi \left(\delta a\right)}{\psi \left(\delta \right)}\underset{0}{\overset{a}{\int }}\left|{\lambda }_{\delta }(a-u)-{\lambda }_{\delta }(a+u)\right|$$

$$+O\left(H(\alpha ,{\tau }_{\delta })\right),$$

we obtain (36). Theorem 4 is proved. □

3. Conclusions

One of the important problems of the approximation theory is studying the approximation properties of linear summation methods of the Fourier series. We have studied the asymptotic behavior of the exact upper bounds of the deviations of operators generated by $\lambda$-methods (defined by the set of functions $\Lambda =\left\{{\lambda }_{\delta }\left(·\right)\right\}$ continuous on $[0;\infty )$ and dependent on the real parameter $\delta$) on the classes of $(\psi ,\beta )$-differentiable functions in O.I. Stepanets sense with $(\psi ,\beta )$-derivatives belonging to the class $H^{\alpha }$. In general, Fourier series summation methods are a powerful tool that can be effectively used to sum Fourier series with high accuracy and stability. The development of this mathematical apparatus can be used in many applied fields. Regarding further research in this field, we should note that similar problems can be considered in the broader classes of functions, such as classes $C_{\beta }^{\psi }{H}_{\omega }$, where $\omega$ is a fixed majorant of the type of modulus continuity.