Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

Absolute tauberian conditions for absolute hausdorff and quasi-hausdorff methods

The purpose of this paper is to prove that for a large set of absolute Hausdorff and quasi-Hausdorff methods the condition $$\sum\limits_{k = 1}^\infty {\left| {\lambda _n a_n - \lambda _{n - 1} a_{n - 1} } \right|< } \infty $$ is a Tauberian condition, i.e., its fulfillment together with the absolute summability of \(\sum\limits_{n = 0}^\infty {a_n } \) tos implies that \(\sum\limits_{n = 0}^\infty {\left| {a_n } \right|}< \infty \) and \(\sum\limits_{n = 0}^\infty {a_n } = s.\) a _n =s.     

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.     

Homogenization of Elasticity Problems with Boundary Conditions of Signorini type

In a perforated domain $\Omega ^\varepsilon = \Omega \cap \varepsilon \omega $ formed of a fixed domain Ω and an ε-compression of a 1-periodic domain ω, we consider problems of elasticity for variational inequalities with boundary conditions of Signorini type on a part of the surface $S_0^\varepsilon $ of perforation. We study the asymptotic behavior of solutions as ε → 0 depending on the structure of the set $S_0^\varepsilon $ . In the general case, the limit (homogenized) problem has the two distinguishing properties: (i) the limit set of admissible displacements is determined by nonlinear restrictions almost everywhere in the domain Ω, i.e., in the limit, the Signorini conditions on the surface $S_0^\varepsilon $ can turn into conditions posed at interior points of Ω (ii) the limit problem is stated for an homogenized Lagrangian which need not coincide with the quadratic form usually determining the homogenized elasticity tensor. Theorems concerning the homogenization of such problems were obtained by the two-scale convergence method. We describe how the limit set of admissible displacements and the homogenized Lagrangian depend on the geometry of the set $S_0^\varepsilon $ on which the Signorini conditions are posed.     

A Criterion for Weak Generalized Localization in the Class L_1 for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function $f \in L_p $ , $p \geqslant 1$ , $f(x) = 0$ , on a set of positive measure $\mathfrak{A} \subset \mathbb{T}^N = [ - \pi ,\pi )^N $ , $N \geqslant 2$ , depending on the rotation of the coordinate system, i.e., depending on the element $\tau \in \mathcal{F}$ , where $\mathcal{F}$ is the rotation group about the origin in $\mathbb{R}^N $ . This problem has been reduced to the study of the change in the geometry of the sets $\tau ^{ - 1} (\mathfrak{A}) \cap \mathbb{T}^N $ (where $\tau ^{ - 1} \in \mathcal{F}$ satisfies $\tau ^{ - 1} \cdot \tau = 1$ ) and $\mathbb{T}^N \backslash {\text{supp}}(f \circ \tau )$ depending on the “rotation,” i.e., on $\tau \in \mathcal{F}$ . In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function $f \circ \tau $ is understood) and give (for both cases) possible solutions of the problem in the class $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

Universelle Approximation durch Riesz-Transformierte der geometrischen Reihe

Let p={p_v} be a fixed sequence of complex numbers. Define \(p_n : = \mathop \Sigma \limits_{\nu = o}^n p_\nu \) and suppose that \(p_{m_k } \ne o\) for a subsequence M={m_k} of nonnegative integers. The matrix A=(α_kv) with the elements $$\alpha _{k\nu } = p_\nu /p_{m_k } if o \leqslant \nu \leqslant m_k ,\alpha _{k\nu } = oif \nu > m_k $$ generates a summability method (R,p,M) which is a refinement of the well known Riesz methods. The (R,p,M) methods have been introduced in [4]. In the present paper we are concerned with the summability of the geometric series \(\mathop \Sigma \limits_{\nu = o}^n z^\nu \) by (R,p,M) methods. We prove the following theorem. Suppose G is a simply connected domain with \(\{ z:|z|< 1\} \subset G,1 \varepsilon | G \) . Then there exists a universal, regular (R,p,M) method having the following properties: (1) \(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is compactly summable (R,p,M) to \(\tfrac{1}{{1 - z}}\) on G. (2) For every compact set B?¯G^c which has a connected complement and for every function f which is continuous on B and analytic in its interior there exists a subsequence M(B,f) of M such that \(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is uniformly summable (R,p,M(B,f)) to f(z) on B. (3) For every open set U?G^c which has simply connected components in ? and for every function f which is analytic on U there exists a subsequence M(U,f) of M such that \(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is compactly summable (R,p,M(U,f)) to f(z) on U.     

Series with monotone coefficients by Walsh system

The paper studies the series $\sum\limits_{n = 0}^\infty {a_n } W_n (x)$ by Walsh system, where |a _n | monotone tends to zero and $\sum\limits_{n = 1}^\infty {a_{_n }^2 } = \infty $ . Some theorems on correction in L ¹ and representability of functions from L ^p , p ∈ (0, 1) by subseries of the Walsh series are proved.     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.     

Banach spaces of locally Schlicht functions with the Hornich operations

Let \(S_ \propto ( \propto \geqq 0)\) be the set of normalized (see (1.2)) functions f holomorphic in D:|z|<1 with \(f''(z)/f'(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and let be the set of normalized (see (1.6)) functions f meromorphic in D with the Schwarzian derivative \(\left\{ {f,z} \right\} = 0((1 - \left| z \right|^2 )^{ - \propto } )\) . We shall show that some topological properties of \(S_ \propto\) and , and of subsets of them, follow from those of the weighted H^∞ space \(H_ \propto ^\infty\) , consisting of functions f holomorphic in D with \(f(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and those of subsets of \(H_ \propto ^\infty\) . The set S₁ is denoted by X in [3] and [4].     

The Wiener--Hopf Integral Equation in the Supercritical Case

We consider the scalar homogeneous equation $S(x) = \int_0^\infty {K(x - t)S(t)dt,{\text{ }}x \in \mathbb{R}^ + \equiv (0,\infty ),}$ with symmetric kernel $K:K( - x) = K(x),{\text{ }}x \in \mathbb{R}_1$ satisfying the conditions $0 \leqslant K \in L_1 (\mathbb{R}^ + ) \cap C^{\left( 2 \right)} (\mathbb{R}^ + )$ , $\int_0^\infty {K(t)dt > \frac{1}{2}} $ , $K' \leqslant 0{\text{ }}and 0 \leqslant K'' \downarrow {\text{ }}on \mathbb{R}^ + $ . We prove the existence of a real solution S of the equation given above with asymptotic behavior $S(x) = O(x){\text{ as }}x \to + \infty $ .     

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.     

Between\widehat{gl}(\infty ) and\widehat{sl}_N affine algebras. I. Geometrical actions

We consider the central extended $\widehat{gl}(\infty )$ Lie algebra and a set of its subalgebras parametrized by |q|=1, which coincides with the embedding of the quantum tori Lie algebras (QTLA) in $\widehat{gl}(\infty )$ . Forq ^N=1 there exists an ideal, and a factor over this ideal is isomorphic to an $\widehat{sl}_{N(z)} $ affine algebra. For a generic valueq the corresponding subalgebras are dense in $\widehat{gl}(\infty )$ . Thus, they interpolate between $\widehat{gl}(\infty )$ and $\widehat{sl}_{N(z)} $ . All these subalgebras are fixed points of automorphism of $\widehat{gl}(\infty )$ . Using the automorphisms, we construct geometrical actions for the subalgebras, starting from the Kirillov-Kostant form and the corresponding geometrical action for $\widehat{gl}(\infty )$ .     

Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series

It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $m \geqslant 2$ and a $2\pi $ -periodic (in each variable) function $f(x) \in C(T^m )$ belongs to the Nikol'skii class $h_\infty ^{(m - 1)/2} (T^m )$ , then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $h_\infty ^{(m - 1)/2} (T^m )$ whose Fourier series is divergent over hyperbolic crosses at some point.     

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.     

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.     

Converse Quadrature sum Inequalities for Freud Weights. ii

Let $W: = \exp \left( { - Q} \right)$ , where $Q$ is of smooth polynomial growth at $\infty$ , for example $Q\left( x \right) = \left| x \right|^\beta ,\beta >1$ . We call $W^2 $ a Freud weight. Let $\left\{ {x_{j{\kern 1pt} n} } \right\}_{j = 1}^n $ and $\left\{ {\lambda _{j{\kern 1pt} n} } \right\}_{j = 1}^n $ denote respectively the zeros of the $n$ th orthonormal polynomial $p_n$ for $W^2 $ and the Christoffel numbers of order $n$ . We establish converse quadrature sum inequalities associated with W, such as $$\left\| {\left( {PW} \right)\left( x \right)\left( {1 + \left| x \right|} \right)^r } \right\|_{L_p \left( R \right)} $$ with $C$ independent of $n$ and polynomials P of degree $ < n$ , and suitable restrictions on $r$ , $R$ . We concentrate on the case ${ \geqq 4}$ , as the case ${p < 4}$ was handled earlier. We are able to treat a general class of Freud weights, whereas our earlier treatment dealt essentially with $\left( { - \left| x \right|^\beta } \right),\beta = 2,4,6,....$ Some applications to Lagrange interpolation are presented.     

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.     

Geometric rigidity for sequences of W^{2,2} conformal immersions

We analyse sequences of discs conformally immersed in $ \mathbb{R }^ n$ with energy $ \int _{ D} |A_k |^ 2 \le \gamma _n$ , where $ \gamma _n = 8\pi $ if $ n=3$ and $ \gamma _n = 4 \pi $ when $n\ge 4$ . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $ n=3$ or Chen’s minimal graph if $ n \ge 4$ . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $ . We apply the above analysis to show that in $ \mathbb{R }^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $ then there exists a sequence of Möbius transforms $ \sigma _{k}$ such that $ \sigma _k\circ f _k$ converges weakly to a catenoid.