Zur Abschätzung der Partitionsfunktion p(n) mit elementaren Mitteln

If p(n) denotes the number of all partitions n=n₁+...+=n_r of a natural number n, the inequality $$(\alpha - \varepsilon )\sqrt n< \log p(n)(with3\alpha = \pi \sqrt 6 )$$ (with 3α=π√6) holds for all n>N(ε). It is shown in the paper, that one can choose $$N(\varepsilon ) = \left( {\frac{{16}}{\varepsilon }} \right)^{\frac{2}{\varepsilon } + 2} $$ The elementary proof, given by GROSSWALD in [2], is modified in such a way, that one needs no results from integral calculus. Furthermore one can see, that the connection of α and π is not essential for the proof. All facts about the constant α, which are needed, can be derived from the formula $$\alpha ^2 = 4 \cdot (1 + \frac{1}{{2^2 }} + \frac{1}{{3^2 }} + ... + \frac{1}{{n^2 }} + ...).$$      

On certain properties of linear summation methods of Fourier series on the classesC(ɛ)

Для линейных методов суммирования рядов Ф урье (1) $$L_n (f;x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2} + \sum\limits_{k = 1}^n {\lambda _{k,n} } \cos kt} \right)dt$$ на классах $$C(\varepsilon ) = \{ f:E_n (f) \leqq \varepsilon _n ;\forall n \geqq 0\} ,\varepsilon = \{ \varepsilon _n \} _{n = 0.}^\infty \varepsilon _n \downarrow 0,$$ доказываются:

1. оценки для порядка р оста норм ∥{L_n∥, если из вестен порядок приближения операторами (1) некоторого классаС (?) (при этом, если опера торы (1) приближают класс С(е) с наилучшим порядком, то находится точная а симптотика возрастания норм {∥ L_n∥);
2. сравнительные оцен ки порядков приближе ния классовС(?) операторами (1), если известен порядок при ближения ими некотор ого более узкого класса С(?*).

В том случае, когда опе раторы (1) приближают кл асс С(?*) с наилучшим порядком, получаются точные по рядковые оценки для л юбого более широкого класса С(?).     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

Optimal rate of integration and ɛ-entropy of a class of analytic functions

The author considers a class F of analytic functions real in the interval [-1, 1] and bounded in the unit circle. As an estimate of the optimal quadrature error R(n) over the class F it is shown that $$_e - \left( {2\sqrt 2 + \frac{1}{{\sqrt 2 }}} \right)\pi \sqrt n \leqslant R(n) \leqslant e^{ - \frac{\pi }{{\sqrt 2 }}n} .$$ With the additional condition that \(\mathop {max}\limits_{x \in [ - 1,1]}\) ¦f(x)¦?B, an estimate is obtained for the ?-entropy H_?(F): $$\frac{8}{{27}}\frac{{(1n2)^2 }}{{\pi ^2 }} \leqslant \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{H_\varepsilon (F)}}{{\left( {\log \frac{1}{\varepsilon }} \right)^3 }} \leqslant \frac{2}{{\pi ^2 }}(1n2)^2 .$$      

Characterization of the Bernoulli-polynomials,exp andΨ by Nörlund's multiplication formula

The system of functional equations $$\forall p\varepsilon N_ + \forall (x,y)\varepsilon D:f(x,y) = \frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f(x + ky,py)}$$ is suited to characterize the functions $$(x,y) \mapsto y^m B_m \left( {\frac{x}{y}} \right),m\varepsilon N,$$ B _m means them-th Bernoulli-polynomial, $$(x,y) \mapsto \exp (x)y(\exp (y) - 1)^{ - 1}$$ (for these functionsD =R ×R ₊) and $$(x,y) \mapsto \log y + \Psi \left( {\frac{x}{y}} \right)(D = R_ + \times R_ + )$$ as those continuous solutions of this system which allow a certain separation of variables and take on some prescribed function values.     

Precise asymptotics of complete moment convergence on moving average

Let {ξi,-∞i∞} be a doubly infinite sequence of identically distributed-mixing random variables with zero means and finite variances,{ai,-∞i∞} be an absolutely summable sequence of real numbers and X k =∑i=-∞+∞ aiξi+k be a moving average process.Under some proper moment conditions,the precise asymptotics are established for     

On a strengthened extremal property of the Carathéodory-Fejér functions

Для заданной на едини чной окружности огра ниченной функцииω(ξ) рассматр ивается усложненная задача а ппроксимации аналит ическими функциями: $$\mathop {\inf }\limits_{\varphi \in H^\infty } \left[ {\left\| {\omega - \varphi } \right\| + \mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k \left| {\lambda _k } \right|} \right],$$ где ∥·∥ понимается вL ^∞,ε _k ≧0 — заданные чис ла, $$\mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k< + \infty ,\varphi (z) = \mathop \Sigma \limits_{k = 0}^\infty \lambda _k z^k .$$ Доказывается, что при всех достаточно малы хε _k экстремальной в этой задаче будет функция обычного наилучшего приближения (та же, что и приε _k =0,k=0, 1, ...). В частности, при $$\omega (\zeta ) = \frac{{\gamma _0 }}{{\zeta ^n }} + \frac{{\gamma _1 }}{{\zeta ^{n - 1} }} + ... + \frac{{\gamma _{n - 1} }}{\zeta }$$ экстремальной оказы вается дробь Каратео дори—Фейера. Переход к двойственн ой задаче позволяет получить т очные оценки для клас са интегралов типа Коши, выделяемого огранич ениями, наложенными на велич ины коэффициентов ря да Тейлора.     

Best approximation and de la Vallée-Poussin sums

For the class C_ε={f∈C_2π: E_n, n≤Z₊} where \(\left\{ {\varepsilon _n } \right\}_{n \in Z_ + } \) is a sequence of numbers tending monotonically to zero, we establish the following precise (in the sense of order) bounds for the error of approximation by de la Vallée-Poussin sums: (1) $$c_1 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \leqslant \mathop {\sup }\limits_{f \in C_\varepsilon } \left\| {f - V_{n, l} \left( f \right)} \right\|_C \leqslant c_2 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \left( {n \in N} \right)$$ , where c₁ and c₂ are constants which do not depend on n orl. This solves the problem posed by S. B. Stechkin at the Conference on Approximation Theory (Bonn, 1976) and permits a unified treatment of many earlier results obtained only for special classes C_ε of (differentiable) functions. The result (1) substantially refines the estimate (see [1]) (2) $$\left\| {V_{n, l} \left( f \right) - f} \right\|_C = O\left( {\log {n \mathord{\left/ {\vphantom {n {\left( {l + 1} \right) + 1}}} \right. \kern-\nulldelimiterspace} {\left( {l + 1} \right) + 1}}} \right) E_n \left[ f \right] \left( {n \to \infty } \right)$$ and includes as particular cases the estimates of approximations by Fejér sums (see [2]) and by Fourier sums (see [3]).     

Large deviations for the empirical mean of an M/M/1 queue

Let $(Q(k):k\ge 0)$ be an $M/M/1$ queue with traffic intensity $\rho \in (0,1).$ Consider the quantity $$\begin{aligned} S_{n}(p)=\frac{1}{n}\sum _{j=1}^{n}Q\left( j\right) ^{p} \end{aligned}$$ for any $p>0.$ The ergodic theorem yields that $S_{n}(p) \rightarrow \mu (p) :=E[Q(\infty )^{p}]$ , where $Q(\infty )$ is geometrically distributed with mean $\rho /(1-\rho ).$ It is known that one can explicitly characterize $I(\varepsilon )>0$ such that $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n}\log P\big (S_{n}(p)<\mu \left( p\right) -\varepsilon \big ) =-I\left( \varepsilon \right) ,\quad \varepsilon >0. \end{aligned}$$ In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n^{1/(1+p)}}\log P\big (S_{n} (p)>\mu \big (p\big )+\varepsilon \big )=-C\big (p\big ) \varepsilon ^{1/(1+p)}, \end{aligned}$$ where $C(p)>0$ is obtained as the solution of a variational problem. We discuss why this phenomenon—Weibullian right tail asymptotics rather than exponential asymptotics—can be expected to occur in more general queueing systems.     

On linear summation methods of Fourier series

Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ _k и их разностейΔλ _k=λ _k?λ _k?1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ?(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<ε≦1/2. ЗдесьS _m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1.     

On averages of Fourier coefficients of Maass cusp forms

Let φ be a primitive Maass cusp form and t _φ (n) be its nth Fourier coefficient at the cusp infinity. In this short note, we are interested in the estimation of the sums ${\sum_{n \leq x}t_{\varphi}(n)}$ and ${\sum_{n \leq x}t_{\varphi}(n^2)}$ . We are able to improve the previous results by showing that for any ${\varepsilon > 0}$ $$\sum_{n \leq x}t_{\varphi}(n) \ll\, _{\varphi, \varepsilon} x^{\frac{1027}{2827} + \varepsilon} \quad {and}\quad\sum_{n \leq x}t_{\varphi}(n^2) \ll\,_{\varphi, \varepsilon} x^{\frac{489}{861} + \varepsilon}.$$      

Well-posedness of a semilinear heat equation with weak initial data

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation $$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$ with initial data in $\dot L_{r,p} $ is studied. We prove the well-posedness when $$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$ and construct non-unique solutions for $$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$ In the second part the well-posedness of the avove IVP for k=2 with μ₀?H ^s (? ⁿ ) is proved if $$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$ and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].     

On semilinear dissipative systems of equations with a small parameter that arise in solution of the Navier-Stokes equations,equation of motion of the Oldroyd fluids,and equations of motion of the Kelvin-Voight fluids

Solutions of the two-dimensional initial boundary-value problem for the Navier-Stokes equations are approximated by solutions of the initial boundary-value problem 9 $$\begin{array}{*{20}c} {\frac{{\partial v}}{{\partial t}}^\varepsilon - v\Delta v^\varepsilon + v_k^\varepsilon v_{x_k }^\varepsilon + \frac{1}{2}v^\varepsilon div v^\varepsilon - \frac{1}{\varepsilon }grad div w^\varepsilon = f_1 ,} \\ {\frac{{\partial w^\varepsilon }}{{\partial t}} + \alpha w^\varepsilon = v^\varepsilon ,} \\ \end{array} $$ 10 $$v^\varepsilon \left| {_{t = 0} = v_0^\varepsilon (x), w^\varepsilon } \right|_{t = 0} = 0, x \in \Omega , v^\varepsilon \left| {_{\partial \Omega } = w^\varepsilon } \right|_{\partial \Omega } = 0, t \in \mathbb{R}^ + $$ . We study the proximity of the solutions of these problems in suitable norms and also the proximity of their minimal global B-attractors. Similar results are valid for two-dimensional equations of motion of the Oldroyd fluids (see Eqs. (38) and (41)) and for three-dimensional equations of motion of the Kelvin-Voight fluids (see Eqs. (39) and (43)). Bibliography: 17 titles.     

Spectral asymptotic behavior of a class of integral operators

Integral operators of the type $$(Tf)(x) = \int_0^1 {\frac{{x^\beta y^\gamma }}{{(x + y)^\alpha }}} f(y)dy,$$ the kernels of which have a singularity at a single point, are discussed. H. Widom's method and some of his results are used to show that, if α>0, β, γ>?1/2, ρ=β+γ?α+1>0, then we have for the distribution function of the singular numbers of the operator, $$\mathop {\lim }\limits_{\varepsilon \to 0} N(\varepsilon ,T)ln^{ - 2} {\textstyle{1 \over \varepsilon }} = {\textstyle{1 \over {2\pi ^2 \varepsilon }}}.$$      

Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.     

The distribution of 4-full numbers

Let Q(x) denote the number of 4-full numbers not exceeding x. It is well known that $$Q(x) = \sum\limits_{j = 4}^7 {r_j x^{1/j} + R(x)}$$ where $$r_j = \mathop {res}\limits_{s = 1/j} (F(s)/s), F(s) = \mathop \prod \limits_P \left( {1 + \frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} \right)$$ and R(x) is the remainder. This paper proves that $$R(x) \ll x^{3626/35461 + \varepsilon }$$ where ε is any positive number.     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Regularized sums of half-integer powers of a Sturm-Liouville operator

We investigate the question of the regularized sums of part of the eigenvalues z_n (lying along a direction) of a Sturm-Liouville operator. The first regularized sum is $$\sum\nolimits_{n = 1}^\infty {(z_n - n - \frac{{c_1 }}{n} + \frac{2}{\pi } \cdot z_n arctg \frac{1}{{z_n }} - \frac{2}{\pi }) = \frac{{B_2 }}{2} - c_1 \cdot \gamma + \int_1^\infty {\left[ {R(z) - \frac{{l_0 }}{{\sqrt z }} - \frac{{l_1 }}{z} - \frac{{l_2 }}{{z\sqrt z }}} \right]} } \sqrt z dz,$$ where the z_n are eigenvalues lying along the positive semi-axis, z _n ² =λ_n, $$l_0 = \frac{\pi }{2}, l_1 = - \frac{1}{2}, l_2 = - \frac{1}{4}\int_0^\pi {q(x) dx,} c_1 = - \frac{2}{\pi }l_2 ,$$ , B₂ is a Bernoulli number, γ is Euler's constant, and \(R(z)\) is the trace of the resolvent of a Sturm-Liouville operator.     

Hayman directions of meromorphic functions

In this paper, we shall prove the existence of the singular directions related to Hayman's problems^[1]. The results are as follows.

1. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: argz= θ₀ (0≤θ₀>2π) such that for every positive ε, every integer p(≠0, ?1) and every finite complex number b(≠0), we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' \cdot \{ f\} ^p = b)} \right\} = + \infty $$
2. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ₀ (0≤θ₀>2π) such that for every positive ε, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$
3. Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition $$\mathop {\lim }\limits_{r \to \infty } \frac{{T(r,f)}}{{(\log r)^3 }} = + \infty $$ then there exists a direction H:z= θ₀ (0≤θ₀>2π) such that for every positive ε, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$

The singular directions in Theorems I–III are called Hayman directions.     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$