On the approximation by de la Vallée Poussin sums and interpolatory polynomials in Lipschitz norms

ПустьM ^m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x _k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$      

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Interior regularity operators

ПустьР - линейный диф ференциальный опера тор с достаточно гладкими коэффициентами. По определению,P явля ется оператором внут ренней регулярности на ω ?R ⁿ т огда и только тогда, когда \(u \in B_{p,k_{ - N} }^{loc} (\Omega )\) и ω′?ω из условия \(Pu \in B_{p,k_s }^{loc} (\Omega ')\) вытекает, что \(u \in B_{p,k_s k}^{loc} (\Omega ')\) , где ?N+1≦s≦N. Соотве тствующий пример: $$Pu = - \Delta u + u c k(\xi ) = \xi _1^2 + \ldots + \xi _n^2 + 1.$$ Указанные операторы характеризуются в ра боте в терминах априорных н еравенств. До^? казывается также сущ ествование локальны х фундаментальных реш ений для оператора, со пряженного кP, а также его гладкос ть вне диагонали. Эти результаты являются аналогами соответствующих рез ультатов для гипоэлл иптических операторов.     

О коэффициентах рядо в Фурье по множествам

The following statement is proved: Theorem.Let f(x), 0≦x≦2π, possess the Fourier expansion $$\mathop \sum \limits_{\kappa = - \infty }^\infty c_\kappa e^{in} \kappa ^x with \bar c_\kappa = c_{ - \kappa } , n_\kappa = - \bar n_{ - \kappa }$$ where {n _k } is a Sidon sequence. Then in order to have $$\mathop \sum \limits_{\kappa = - \infty }^\infty |c_\kappa |^p< \infty$$ for a given p, 1 $$\mathop \sum \limits_{k = 1}^\infty \left( {\frac{{\left\| f \right\|L^k (0,2\pi )}}{k}} \right)^p< \infty$$ . An analogous statement holds true for series with respect to the Rademacher system.     

On chagnes of signs in infinite series

Доказана теорема. Пус ть {а_n} — последовате льность положительн ых чисел, монотонно стре мящаяся к нулю, причем Σa _n =∞. Тогда найд ется такая последовательность знаковε(п)=±1 (п=1,2, ...), что для каждого натураль ного т и любого целого b с условием 0≦b≦m?1 выполнено равен ство $$\mathop \sum \limits_{n \equiv b(mod m)} \varepsilon (n)a_n = 0$$ . Приведены также неко торые родственные ре зультаты и указаны нерешенные в опросы.     

Elliptische Differentialgleichungen mit variablen Koeffizienten in Gebieten mit unbeschränktem Rand

We discuss the spectrum of a symmetric elliptic differential operator A with domain \(\mathop {H^m }\limits^o (\Omega ) \cap H^{2m} (\Omega )\) in regions Ω with unbounded boundary \(\dot \Omega \) , where are \(\bar \Omega \) uniformely of class C^2m and on \(\dot \Omega \) the normal condition x·ν(x)≦μ for sufficient small positiveμ. We prove the A-priori-estimate \(\parallel u\parallel _{m,\Omega } \leqq c\parallel (l + r) (A - k)u\parallel _{o,\Omega } \) and show for all k>k, k≧0 suitable, there are no eigenvalues of A and by characterizing weighted Sobolev spaces with negative norm the existence of solutions \((l + r)_2 ^{ - 1} u \in \mathop H\limits^0{^m} (\Omega ) \cap H^{2m} (\Omega )\) of the equation (A?k)u=f, (1+r)f∈L²(Ω).     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

Fixed points and functional equations connected with derivations on Banach algebras

Let 1≦m≦4 be a fixed integer and let f:X→Y be a mapping with X, Y two vector spaces. The functional equation (1.1) is said to be additive if m=1, quadratic if m=2, cubic if m=3 and quartic if m=4, respectively. For convenience, a solution of (1.1) will be called an m-mapping. Let $\mathcal{A}$ , $\mathcal{B}$ be two algebras. An m-mapping $f\colon \mathcal{A} \to \mathcal{B}$ will be called m-derivation if the equality f(xy)=x ^m f(y)+f(x)y ^m is fulfilled for all $x,y\in \mathcal{A}$ . In this paper, we use a fixed point method to prove the stability and hyperstability of m-derivations on Banach algebras.     

Integrability of vector-valued lacunary series with applications to function spaces

Let ${\mathcal L(r) = \sum_{n=0}^\infty a_nr^{\lambda_n}}$ be a lacunary series converging for 0 <  r < 1, with coefficients in a quasinormed space. It is proved that $$\int_0^1 F(1-r,\|\mathcal L(r)\|)(1-r)^{-1}\,{\rm d}r < \infty $$ if and only if $$ \sum_{n=0}^\infty F(1/\lambda_n,\|a_n\|) < \infty, $$ where F is a “normal function” of two variables. In the case when p ≥ 1 and F(x, y) =  x y ^p , this reduces to a theorem of Gurariy and Matsaev. As an application we prove that if ${f(r\zeta) = \sum_{n=0}^\infty r^{\lambda_n}f_{\lambda_n}(\zeta)}$ is a function harmonic in the unit ball of ${\mathbb R^N,}$ then $$\int_0^1M_p^q(r,f)(1-r)^{q\alpha-1} \,{\rm d}r <\infty\quad (p,\,q,\,\alpha >0 ) $$ if and only if $$\sum_{n=0}^\infty \|f_{\lambda_n} \|^q_{L^p(\partial B_N)}(1/\lambda_n)^{q\alpha} <\infty. $$      

Lacunary subsets of orthonormal sets

В РАБОтЕ РАссМАтРИВА УтсьS _Р-пОДсИстЕМы О. Н.с. В ЧАстНОстИ, ДОкАжыВА Етсь слЕДУУЩАь тЕОРЕ МА, кОтОРАь НЕУсИльЕМА. тЕОРЕМА.пУсть Р>2 —ЧЕ тНОЕ ЧИслО, δ — пРОИжВО льНОЕ ЧИслО, 0<δ≦p?2,Φ= {Φ _n(x)} _n=1 ^N —O.H.C.,x?[0,1],пРИЧЕМ ∥ Φ _n∥_p≦M, n=1,2,...,N, гДЕР=Р+δ, 0М<∞. тОгДА Иж сИстЕМы Ф МОж НО ВыБРАть пОДсИстЕМ У \(\Phi ' = \left\{ {\varphi _{n_k } } \right\}_{k = 1}^{N'} ,N' \geqq N^{\alpha (\delta )} ,\alpha (\delta ) = \frac{{2\delta }}{{p(p - 2 + \delta )}}\) , тАкУУ, ЧтО Дль лУБОгО п ОлИНОМА \(P(x) = \sum\limits_{k = 1}^{N'} {a_k \varphi _{n_k } (x)} \) ИМЕЕ т МЕстО ОцЕНкА $$(\mathop \sum \limits_{k = 1}^{{\rm N}'} a_k^2 )^{1/2} \leqq \left\| P \right\|_p \leqq c_{p,M,\delta } (\mathop \sum \limits_{k = 1}^{{\rm N}'} a_k^2 )^{1/2} $$ (c _p, m, δ — пОстОьННАь, жАВИ сьЩАь тОлькО Отp, M, δ, НО НЕ От N ИлИ кОЁФФИцИЕНтОВ пО лИ-НОМА). пРИВОДьтсь И ДРУгИЕ РЕжУльтАты А НАлОгИЧНОгО хАРАктЕ РА.     

A tauberian theorem for random walk

LetX ₁,X ₂,... be independent random variables, all with the same distribution symmetric about 0; $$S_n = \sum\limits_{i = 1}^n {X_i } $$ It is shown that if for some fixed intervalI, constant 1<a≦2 and slowly varying functionM one has $$\sum\limits_{k = 1}^n {P\{ S_k \in I\} \sim \frac{{n^{1 - 1/\alpha } }}{{M(n)}}} (n \to \infty )$$ then theX _i belong to the domain of attraction of a symmetric stable law.     

Traces, traceability, and lattices of traces under the set theoretic inclusion

Let a trace be a computably enumerable set of natural numbers such that ${V^{[m]} = \{n : \langle n, m\rangle \in V \}}$ V [ m ] = { n : 〈 n , m 〉 ∈ V } is finite for all m, where ${\langle^{.},^{.}\rangle}$ 〈 . , . 〉 denotes an appropriate pairing function. After looking at some basic properties of traces like that there is no uniform enumeration of all traces, we prove varied results on traceability and variants thereof, where a function ${f : \mathbb{N} \rightarrow \mathbb{N}}$ f : N → N is traceable via a trace V if for all ${m, \langle f(m), m\rangle \in V.}$ m , 〈 f ( m ) , m 〉 ∈ V . Then we turn to lattices $$\textit{\textbf{L}}_{tr}(V) = (\{W : V \subseteq W \, {\rm and} \, W \, {\rm a} \, {\rm trace}\}, \, \subseteq),$$ L t r ( V ) = ( { W : V ? W and W a trace } , ? ) , V a trace. Here, we study the close relationship to ${\mathcal{E} = (\{A : A \subseteq \mathbb{N} \quad c.e.\}, \subseteq)}$ E = ( { A : A ? N c . e . } , ? ) , automorphisms, isomorphisms, and isomorphic embeddings.     

An integral equation and a representation for a Green's function

The final step in the mathematical solution of many problems in mathematical physics and engineering is the solution of a linear, two-point boundary-value problem such as $$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< x \hfill \\ (0) = 0, \dot u(x) = 0 \hfill \\ \end{gathered} $$ Such problems frequently arise in a variational context. In terms of the Green's functionG, the solution is $$u(t) = \int_0^x {G(t, y, x)g(y) dy} $$ It is shown that the Green's function may be represented in the form $$G(t,y,x) = m(t,y) - \int_y^x {q(s)m(t, s) m(y, s)} ds, 0< t< y< x$$ wherem satisfies the Fredholm integral equation $$m(t,x) = k(t,x) - \int_0^x k (t,y) q(y) m(y, x) dy, 0< t< x$$ and the kernelk is $$k(t, y) = min(t, y)$$      

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case

We consider the generalized Gagliardo-Nirenberg inequality in $\Bbb{R}^{n}$ including homogeneous Besov space $\dot{B}^{s}_{r,\rho}(\Bbb{R}^{n})$ with the critical order s=n/r, which describes the continuous embedding such as $L^{p}(\Bbb{R}^{n})\cap\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})\subset L^{q}(\Bbb{R}^{n})$ for all q with p ≦ q<∞, where 1≦ p ≦ r<∞ and 1<ρ ≦∞. Indeed, the following inequality holds: $$\|u\|_{L^{q}(\Bbb{R}^{n})}\leqq C\,q^{1-1/\rho}\|u\|_{L^{p}(\Bbb{R}^{n})}^{p/q}\|u\|_{\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})}^{1-p/q},$$ where C is a constant depending only on r. In this inequality, we have the exact order 1?1/ρ of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space $\dot{F}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ , which implies the usual Sobolev imbedding with the critical Sobolev space $\dot{H}^{n/r}_{r}(\Bbb{R}^{n})$ . Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in $\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ .     

Maxima for the Expectation of the Lifetime of a Brownian Motion in the Ball

Let G _B (x, y) be the Green’s function of the unit ball B in ${\mathbb{R}^n, n \ge 3,}$ and ${\Gamma_B (x,y)=\int_BG_B(x, z)G_B(z, y)dz}$ the iterated Green’s function. The function $$E_x^y(\tau_B) = \frac{\Gamma_B(x, y)}{G_B(x, y)}$$ is the expectation of the lifetime of a Brownian motion starting at ${x \in \overline{B}}$ , killed on exiting B and conditioned to converge to and to be stopped at ${y \in \overline{B}}$ . The aim of the paper is to prove that $$\sup_{x \in \partial B,y \in B} E_x^y(\tau_B) = \sup_{x,y \in \partial B} E_x^y(\tau_B) = E_{x_0}^{-x_0}(\tau_B), x_0 \in\partial B$$ and that the maximum value of ${E_x^y(\tau_B)}$ occurs if and only if x, y are diametrically opposite points on the boundary of B.     

On initial segments of degrees of constructibility

Let \(\mathfrak{M}\) be a fixed countable standard transitive model of ZF+V=L. We consider the structure Mod of degrees of constructibility of real numbers x with respect to \(\mathfrak{M}\) such that \(\mathfrak{M}\) (x) is a model. An initial segment Q \( \subseteq \) Mod is called realizable if some extension of \(\mathfrak{M}\) with the same ordinals contains exclusively the degrees of constructibility of real numbers from Q (and is a model of Z FC). We prove the following: if Q is a realizable initial segment, then $$[y \in Q \to y< x]]\& \forall z\exists y[z< x \to y \in Q\& \sim [y< z]]]$$ .     

Infinitely many solutions for the prescribed curvature problem of polyharmonic operator

We consider the following prescribed curvature problem for polyharmonic operator: $$\left\{\begin{array}{llll} D_{m} u = \tilde{K}(y)|u|^{m^*-2}u\; {\rm in}\; \mathbb{S}^N\\ u \quad\; >0\qquad\quad\quad\quad\quad{\rm on}\; \mathbb{S}^N\\ u \quad\; \in H^{m}(\mathbb{S}^N), \end{array} \right.$$ where ${m^*=\frac{2N}{N-2m}, N\geq 2m+1,m \in \mathbb{N}_{+}, \tilde{K}}$ is positive and rationally symmetric, ${\mathbb{S}^N}$ is the unit sphere with the induced Riemannian metric ${g=g_{\mathbb{S}^N},}$ and D _m is the elliptic differential operator of 2m order given by $$\begin{array}{lll}D_m={\prod\limits_{k=1}^m}{\left(-\Delta_g+\frac{1}{4}(N-2k)(N+2k-2)\right)}\end{array}$$ where Δ _g is the Laplace-Beltrami operator on ${\mathbb{S}^N}$ . We will show that problem (P) has infinitely many non-radial positive solutions, whose energy can be arbitrary large.