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.     

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.     

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.     

Oscillation theorems for second-order nonlinear delay difference equations

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation $$\Delta (p_n (\Delta x_n )^\gamma ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;\;n = 0,1,2,...,$$ when $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ . When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} < \infty }$ we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.     

Theory of nonstationary flows of Kelvin-Voigt fluids

One proves the global unique solvability in class \(W_\infty ^1 (0,T;C^{2,d} (\bar \Omega ) \cap H(\Omega ))\) of the initial-boundary-value problem for the quasilinear system $$\frac{{\partial \vec \upsilon }}{{\partial t}} + \upsilon _k \frac{{\partial \vec \upsilon }}{{\partial x_k }} - \mu _1 \frac{{\partial \Delta \vec \upsilon }}{{\partial t}} - \int\limits_0^t {K(t - \tau )\Delta \vec \upsilon (\tau )d\tau + grad p = \vec f,di\upsilon \bar \upsilon = 0,\upsilon , > 0.}$$ This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation $$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$      

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

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

Estimation of the remainder of a cubature formula on a Chebyshev grid for two-variable functions

For a cubature formula of the form $$\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {f(x,y)dxdy = \frac{{4\pi ^2 }} {{mn}}\sum\limits_{i = 0}^{n - 1} {\sum\limits_{j = 0}^{m - 1} {f\left( {\frac{{2\pi i}} {n},\frac{{2\pi j}} {m}} \right) + R_{n,m} (f)} } } }$$ on a Chebyshev grid, the remainder R _n,m (f) is proved to satisfy the sharp estimate $$\mathop {\sup }\limits_{f \in H\left( {r_1 ,r_2 } \right)} \left| {R_{n,m} (f)} \right| = O\left( {n^{ - r_1 + 1} + m^{ - r_1 + 1} } \right)$$ in some class of functions H(r ₁, r ₂) defined by a generalized shift operator. Here, r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in the O-term depends only on ??.     

Inequalities for the Volume of the Unit Ball in {\mathbb{R}^{n}}

The volume of the unit ball in ${\mathbb{R}^{n}}$ is defined by $$\Omega_{n} = \frac{\pi^{n/2}}{\Gamma(n/2+1)},\qquad n = 1,2,3,\ldots,$$ where Γ denotes the classical gamma function of Euler. In several recently published papers numerous authors studied properties of Ω _n . In particular, various inequalities involving Ω _n are given in the literature. In this paper, we continue the work on this subject and offer new inequalities. More precisely, we offer sharp upper and lower bounds for $$\frac{\Omega_{n}^{2}}{\Omega_{n-1} \Omega_{n+1}},\quad\frac{\Omega_{n}}{\Omega_{n-1}+\Omega_{n+1}} \quad {\rm and} \quad\Omega_{n}.$$      

The number of strictly increasing mappings of fences

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a _n,m+b _n,m+c _n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

On some new identities for the Fibonomial coefficients

Let F _n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.     

Some properties of elliptic Riesz means at critical index on Hp(TH)

Suppose f∈H^p(Tⁿ), 0 _r ^δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.     

On the divergence of Lagrange interpolation processes on a set of second category

В статье даны полные д оказательства следу ющих утверждений. Пустьω — непрерывная неубывающая полуадд итивная функций на [0, ∞),ω(0)=0 и пусть M?[0, 1] — матрица узл ов интерполирования. Если $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n > 0$$ то существует точкаx ₀∈[0,1] и функцияf ∈ С[0,1] таки е, чтоω(f, δ)=О(ω(δ)), для которой $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x_0 ) - f(x_0 )| > 0$$ Если же $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n = \infty$$ , то существуют множес твоE второй категори и и функцияf ∈ С[0,1],ω(f, δ)=o(ω(δ)) та кие, что для всехx∈E $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x)| = \infty$$ . Исправлена погрешно сть, допущенная автор ом в [5], и отмеченная в работе П. Вертеши [9].     

Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

Volume Formulas in Lp-Spaces

We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments $${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$ where x1,...,xm are m vectors in $\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1$ . According to the definition of Firey, the Minkowski p-sum of segments is given by $$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$ We describe related geometric properties of the Lewis maps associated to classical operator norms.     

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.     

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σ _d|n d. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.     

On oscillatory integral Hilbert transformation with trigonometric polynomial phase

Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C _n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a _k , b _k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C _n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial.     

Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces L p [0, 1]

We obtain conditions for the convergence in the spaces L ^p [0, 1], 1 ≤ p < ∞, of biorthogonal series of the form $$ f = \sum\limits_{n = 0}^\infty {(f,\psi _n )\phi _n } $$ in the system {? _n } _n≥0 of contractions and translations of a function ?. The proposed conditions are stated with regard to the fact that the functions belong to the space $ \mathfrak{L}^p $ of absolutely bundleconvergent Fourier-Haar series with norm $$ \left\| f \right\|_p^ * = \left| {f,\chi _0 } \right| + \sum\limits_{k = 0}^\infty {2^{k({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})} } \left( {\sum\limits_{n = 2^k }^{2^{k + 1} - 1} {\left| {f,\chi _n } \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where (f,χ _n ), n = 0, 1, ..., are the Fourier coefficients of a function f ? L ^p [0, 1] in the Haar system {χ _n } _n≥0. In particular, we present conditions for the system {? _n } _n≥0 of contractions and translations of a function ? to be a basis for the spaces L ^p [0, 1] and $ \mathfrak{L}^p $ .