Approximating periodic functions in Hölder type metrics by singular integrals with positive Kernels

Let M be either the space of 2π-periodic functions L_p, where 1 ≤ p < ∞, or C; let ω_r(f, h) be the continuity modulus of order r of the function f, and let

Uniform estimates of positive solutions to quasi-linear differential equations

One considers the differential inequality

On a multiplicative type sum form functional equation and its role in information theory

In this paper, we obtain all possible general solutions of the sum form functional equations

Congruences involving the Fermat quotient

Let p > 3 be a prime, and let q _p (2) = (2 ^p?1 ? 1)/p be the Fermat quotient of p to base 2. In this note we prove that $$\sum\limits_{k = 1}^{p - 1} {\frac{1}{{k \cdot {2^k}}}} \equiv {q_p}(2) - \frac{{p{q_p}{{(2)}^2}}}{2} + \frac{{{p^2}{q_p}{{(2)}^3}}}{3} - \frac{7}{{48}}{p^2}{B_{p - 3}}(\bmod {p^3})$$ , which is a generalization of a congruence due to Z.H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z.H. Sun, we show that $${q_p}{(2)^3} \equiv - 3\sum\limits_{k = 1}^{p - 1} {\frac{{{2^k}}}{{{k^3}}}} + \frac{7}{{16}}\sum\limits_{k = 1}^{(p - 1)/2} {\frac{1}{{{k^3}}}} (\bmod p)$$ , which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum\limits_{k = 1}^{p - 1} {{1 \mathord{\left/ {\vphantom {1 {\left( {k^2 \cdot 2^k } \right)}}} \right. \kern-0em} {\left( {k^2 \cdot 2^k } \right)}}}$ modulo p ² that also generalizes a related Sun’s congruence modulo p.     

Construction of a solution of a mixed problem for the generalized heat conduction equation by the fourier method

By the Fourier method a solution of the equation

Inequalities for absolute maxima of a polynomial and its derivative

Suppose z ₁, z ₂, ... z _n are complex numbers with absolute values more than 1 and Arg z _j Arg z _k for j k where Arg w stands for the argument of the complex number w in [0,2). In this note we show that

The area of an exponential random walk and partial sums of uniform order statistics

Let S_i be a random walk with standard exponential increments. The sum ∑ _i=1 ^k S_i is called the k-step area of the walk. The random variable ∑ _i=1 ^k S_i plays an important role in the study of the so-called one-dimensional sticky particles model. We find the distribution of this variable and prove that

Series and product representations for some mathematical constants

We present several series and product representations for γ, π, and other mathematical constants. One of our results states that, for all real numbers μ s>0, we have

On the divergence of Nörlund logarithmic means of Walsh-Fourier series

It is well known in the literature that the logarithmic means 1/logn ^n-1∑k=1 Sk（f）/k of Walsh or trigonometric Fourier series converge a.e. to the function for each integrable function on the unit interval. This is not the case if we take the partial sums. In this paper we prove that the behavior of the so-called NSrlund logarithmic means 1/logn ^n-1∑k=1 Sk（f）/n-k is closer to the properties of partial sums in this point of view.     

Uniformly maldistributed sequences in a strict sense

It is shown that the following three limits

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form

Oscillation for systems of nonlinear neutral type parabolic partial functional differential equations

This paper discusses the oscillation of solutions for systems of nonlinear neutral type parabolic partial fuctional differential equations of the form     

Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups

The (Nörlund) logarithmic means of the Fourier series is:

$t_n f = \frac{1}{{l_n }}\sum\limits_{k = 1}^{n - 1} {\frac{{S_k f}}{{n - k}}} , where l_n = \sum\limits_{k = 1}^{n - 1} {\frac{1}{k}} $

. In general, the Fejér (C,1) means have better properties than the logarithmic ones. We compare them and show that in the case of some unbounded Vilenkin systems the situation changes.

     

On a separating solution of a recurrent equation

We consider the recurrent equation

Use of complex analysis for deriving lower bounds for trigonometric polynomials

It is shown that for any distinct natural numbersk ₁,...,k _n and arbitrary real numbersa ₁,...,a _n the following inequality holds:

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .     

General maximal inequalities related to the strong law of large numbers

For any sequence (ξ _n ) of random variables, we obtain maximal inequalities from which we can derive conditions for the a.s. convergence to zero of the normalized differences $$\frac{1}{{2^n }}\left( {\mathop {\max }\limits_{2^n \leqslant k < 2^{n + 1} } \left| {\sum\limits_{i = 2^n }^k {\xi _i } } \right| - \left| {\sum\limits_{i = 2^n }^{2^{n + 1} - 1} {\xi _i } } \right|} \right).$$ The convergence to zero of this sequence leads to the strong law of large numbers (SLLN). In the special case of quasistationary sequences, we obtain a sufficient condition for the SLLN; this condition is an improvement on the well-known Móricz conditions.     

Approximation of several dimensional functions by trigonometric polynomials

Let f(x, y) be a periodic function defined on the region D

On the extra strong approximation of orthogonal series

В РАБОтЕ УлУЧшЕНы НЕк ОтОРыЕ РЕжУльтАты ОБ ОЧЕНь сИльНОИ И ЁкстРА сИль НОИ АппРОксИМАцИИ. гРУБО гОВОРь, НОВыЕ РЕ жУльтАты сОДЕРжАт тЕ жЕ УтВЕРжДЕНИь, ЧтО И В БО лЕЕ РАННИх тЕОРЕМАх, НО пРИ МЕНЕЕ ОгРАНИЧИтЕльНых Усл ОВИьх НА пАРАМЕтРы. ОсНОВНОИ Р ЕжУльтАт сОстОИт В слЕДУУЩЕМ: ЕслИ γ>0, 00, 0 $$\left\{ {\frac{1}{{n + 1}}\sum\limits_{v = 0}^n {\left| {S_{k_v } (x) - f(x)} \right|^k } } \right\}^{1/k} = o_x (n^{ - y} )$$ пОЧтИ ВсУДУ НА (a,b) Дль кА жДОИ ВОжРАстАУЩЕИ пОслЕДОВАтЕльНОстИ {k _v }.