On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

On simultaneous approximation by lagrange interpolating polynomials

This paper considers to replace △_m(x)=(1-x~2)~2(1/2)/n +1/n~2 in the following result for simultaneousLagrange interpolating approximation with (1-x~2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then|f~(k)(x)-P_~(k)(f,x)|=O(1)△_(n)~(a-k)(x)ω(f~(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q,where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_nU Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.     

On Certain Integral Operators of Fractional Type

In this paper we study integral operators of the form $$T\,f\left( x \right) = \int {k_1 \left( {x - a_1 y} \right)k_2 \left( {x - a_2 y} \right)...k_m \left( {x - a_m y} \right)f\left( y \right)dy} ,$$ $$k_i \left( y \right) = \sum\limits_{j \in Z} {2^{\frac{{jn}}{{q_i }}} } \varphi _{i,j} \left( {2^j y} \right),\,1 \leqq q_i < \infty ,\frac{1}{{q_1 }} + \frac{1}{{q_2 }} + ... + \frac{1}{{q_m }} = 1 - r,$$ $0 \leqq r < 1$ , and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:L^p \left( {R^n } \right) \to T:L^q \left( {R^n } \right)$ for $1 < p < \frac{1}{r}$ and $\frac{1}{q} = \frac{1}{p} - r$ .     

A conjecture of Segre on diophantine approximation

In 1945,B. Segre proved the following classical theorem: Every irrational ξ has an infinity of rational approximationsp/q such that (0) $$\frac{{ - 1}}{{q^2 \sqrt {1 + 4\tau } }}< \frac{p}{q} - \xi< \frac{\tau }{{q^2 \sqrt {1 + 4\tau } }},$$ where τ is any given non-negative real number. Segre conjectured that when τ≠0 and τ^?1 is not an integer, inequalities (0) can be improved by replacing \(\sqrt {1 + 4\tau } \) and \(\sqrt {1 + 4\tau } /\tau \) with larger numbers. In this paper we prove that these two numbers can be replaced with the larger numbers \(\sqrt {1 + 4\tau } + 0.2\tau ^2 \{ \tau ^{ - 1} \} (1 - \{ \tau ^{ - 1} \} )\) and \(\sqrt {1 + 4\tau } /\tau + 0.2\tau ^2 \{ \tau ^{ - 1} \} (1 - \{ \tau ^{ - 1} \} )\) respectively, where {τ^?1} is the fractional part of τ^?1.     

Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D \ne d^2$ and ${{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \right. \kern-0em} {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right),$ where $\sigma _f (n)$ is the singular series, $c(\theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.     

On the difference equationx_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - 1}^p }}

We study, firstly, the dynamics of the difference equation $x_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - 1}^p }}$ , withp ∈ (0,1) and α∈ [0, ∞). Then, we generalize our results to the (k + 1)th order difference equation $x_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - k}^p }}$ ,k = 2, 3,... with positive initial conditions.     

A Theorem on the Zeros of Entire Functions and Its Application

We consider entire functions of exponential type ≤ σ that are bounded and real on $\mathbb{R}$ and satisfy the estimate $( - 1)^k f(k\pi /\sigma + \tau ) \geqslant 0, k \in \mathbb{Z}$ . It is proved that the zeros of such functions are real and simple with the possible exception of points of the form $k\pi /\sigma + \tau $ , which can be zeros of multiplicity at most 2. These results are applied to specific classes of functions and to the problem of the stability of entire functions. We also refine and supplement a few results due to Pólya.     

Asymptotic behavior of multiperiodic functionsG(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )}

Let 0≤g be a dyadic Hölder continuous function with period 1 and g(0)=1, and let $G(x) = \prod\nolimits_{n = 0}^\infty {g(x/{\text{2}}^n )} $ . In this article we investigate the asymptotic behavior of $\smallint _0^{\rm T} \left| {G(x)} \right|^q dx$ and $\frac{1}{n}\sum\nolimits_{k = 0}^n {\log g(2^k x)} $ using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions.     

New estimates of continuity in $$M/GI/1/ \infty$$ queues

The paper deals with an estimation of the total variation distance between stationary distributions of waiting time in two queueing systems with equal Poisson inputs and different distributions B and $\widetilde B$ of service time. Assuming equality of two first moments of B and $\widetilde B$ the continuity inequalities are derived in terms of difference pseudomoments of B and $\widetilde B$ . When in addition the third moments of B and $\widetilde B$ coincide then the constant involved in the corresponding inequality has the asymptotics ${\text{O}}\left[ {\left( {1 - \rho } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} } \right]$ in the heavy traffic limit $\rho \to 1$ .     

Arrangements of Hyperplanes and the Number of Threshold Functions

Two approaches on estimating the number of threshold functions which were recently developed by the author are discussed. Let P(K,n) denote the number of threshold functions in K-valued logic. The first approach establishes that $$P(K,n + 1) \geqslant \frac{1}{2}\left( {\mathop {K^{n - 1} }\limits_{\left\lfloor {n - 4 - 2\frac{n}{{\log _K n}}} \right\rfloor } } \right)P\left( {K,\left\lfloor {{\text{2}}\frac{n}{{\log _K n}} + 3} \right\rfloor } \right).$$ The key argument of investigation is the generalization of the result of Odlyzko on subspaces spanned by random selections of ±1-vectors. Let $E_K = \{ 0,1 \ldots ,K - 1\} $ and let E denote the set of all vectors $w_i ,i = 1, \ldots ,K^n $ , which have the form $(1,a_1 , \ldots ,a_n ),a_i \in E_K $ . Denote by $\Lambda _n (K)$ the number of all collections of different vectors $(w_{i_1 } , \ldots ,w_{i_n } ),2 \leqslant i_1 , \ldots ,i_n \leqslant \mathbb{K}^n $ , such that, for any k, $1 \leqslant k \leqslant n$ , the vector $w_{i_k } $ is minimal among all vectors from the set $E \cap {\text{span}}(w_{i_k } , \ldots ,w_{i_n } )$ . The second approach is based on topology-combinatorical techniques and allows to establish the following inequality $P(K,n) \geqslant 2\Lambda _n (K)$ .     

The Concentration Function of Additive Functions with Special Weight

Suppose that $${g\left( n \right)}$$ is an additive real-valued function, W(N) = 4+ $$\mathop {\min }\limits_\lambda $$ ( λ₂ + $$\sum\limits_{p < N} {\frac{1}{2}} $$ min (1, ( g(p) - λlog p)²), E(N) = 4+1 $$\sum\limits_{\mathop {p < N,}\limits_{g(p) \ne 0} } {\frac{1}{p}.} $$ In this paper, we prove the existence of constants C₁, C₂ such that the following inequalities hold: $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) \in [a,a + 1)} \right\}} \right| \leqslant \frac{{C_1 N}}{{\sqrt {W\left( N \right)} }},$ $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) = a} \right\}} \right| \leqslant \frac{{C_2 N}}{{\sqrt {E\left( N \right)} }},$ . The obtained estimates are order-sharp.     

On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation

In this paper, we study the behavior of the Fourier--Haar coefficients $a_{m_1 , \ldots ,m_n } \left( f \right)$ of functions $f$ Lebesgue integrable on the $n$ -dimensional cube $D_n = \left[ {0,1} \right]^n $ and having a bounded Vitali variation $V_{D_n } f$ on it. It is proved that $$\sum\limits_{m_1 = 2}^\infty \cdots \sum\limits_{m_n = 2}^\infty {\left| {a_{m_1 , \ldots ,m_n } \left( f \right)} \right|} \leqslant \left( {\frac{{2 + \sqrt 2 }}{3}} \right)^n {\text{ }}.{\text{ }}V_{D_n } f$$ and shown that this estimate holds for some function of bounded finite nonzero Vitali variation.     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.     

Sensitivity of best recovery in the Sobolev spaces W_\infty ^{r,d} , \widetilde W_\infty ^{r,d}for perturbed sampling

Let $I^d $ be the d‐dimensional cube, $I^d = [0,1]^d $ , and let $F \ni f \mapsto Sf \in L_\infty (I^d ) $ be a linear operator acting on the Sobolev space F, where Fis either $$$$ or $$$$ where $$\left\| f \right\|_F = \sum\limits_{\left| m \right| = r} {\mathop {{\text{esssup}}}\limits_{x \in I^d } \left| {\frac{{\partial f^{\left| m \right|} }} {{\partial x_1^{m_1 } \partial x_2^{m_2 } \cdot \cdot \cdot \partial x_d^{m_d } }}(x)} \right|.} $$ We assume that the problem elements fsatisfy the condition $\sum\nolimits_{\left| m \right| = r} {{\text{esssup}}} _{x \in I^d } \left| {f^{(m)} (x)} \right| \leqslant 1 $ and that Sis continuous with respect to the supremum norm. We study sensitivity of optimal recovery of Sfrom inexact samples of ftaken at npoints forming a uniform grid on $I^d $ . We assume that the inaccuracy in reading the sample vector is measured in the pth norm and bounded by a nonnegative number δ. The sensitivity is defined by the difference between the optimal errors corresponding to the exact and perturbed readings, respectively. Our main result is that this difference is bounded by $\mathcal{A}\delta $ , where $\mathcal{A} $ is a positive constant independent of the number of samples. This indicates that the curse of dimension, which badly affects the optimal errors, does not extend to sensitivity.     

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 $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

On Total Incomparability of Mixed Tsirelson Spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$ with index $i\left( {\mathcal{M}_k } \right)$ finite are either c ₀ or $\ell _p $ saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left( {\mathcal{M}_k } \right)$ and the parameter θ _k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$ in terms of the asymptotic behaviour of the sequence $\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$ where (e _i is the canonical basis.     

Exact Non-Self-Similar Solutions of the Equation $$u_t = \Delta \ln u$$

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.     

Best Uniform Rational Approximations of Functions by Orthoprojections

Let C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$      

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .