О последовательност ях сумм Фурье-Уолша ограниченных функци й

Let L^∞ denote the space of measurable 1-periodic essentially bounded functionsf(x) with ∥f∥=vrai sup ¦f(x)¦,S _k (f, x) thek-th partial sum of the Walsh-Fourier series off(x),L _k thek-th Lebesgue constant. The following theorem is proved. Theorem. Letλ={λ _K } be a sequence of nonnegative numbers, $$\left\| \lambda \right\|_1 = \mathop \sum \limits_{k = 1}^\infty \lambda _k< \infty ,\left\| \lambda \right\|_2 = (\mathop \sum \limits_{k = 1}^\infty \lambda _k^2 )^{1/2} ,m = log[(\left\| \lambda \right\|_1 /\left\| \lambda \right\|_2 )]$$ .Then for an arbitrary function f∈L ^∞ the following inequalities hold true $$\begin{gathered} \left\| {\mathop \sum \limits_{k = 1}^\infty \lambda _k \left| {S_k (f,x)} \right|} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - 2m]} + c)\left\| f \right\|, \hfill \\ \hfill \\ \mathop \sum \limits_{k = 1}^\infty \lambda _k \left\| {S_k (f)} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - m]} + c)\left\| f \right\| \hfill \\ \end{gathered} $$ , where[y] denotes integral part of a number y>0 and c is an absolute constant. A corollary of the above theorem is that for each functionfεL ^∞ the Lebesgue estimate can be refined for a certain sequence of indices, while the growth order of Lebesgue constants along that sequence can be arbitrarily close to the logarithmic one. “In the mean”, however, the Lebesgue estimate is exact. A further corollary deals with strong summability.     

О стРУктУРЕ цЕлОИ ФУН кцИИ, ОБРАЩАУЩЕИсь В НУль Н А жАДАННОИ пОслЕДОВАтЕльНОстИ тОЧЕк

The following result is proved. Theorem.Let λ _n ,0<λ _n ↑∞, be a sequence of positive numbers with finite density $$\sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ and let a compact set K has the following property: it intersects the real axis along the interval [a, b], where a is the very left point of K, B is the very right point of K; furthermore, K intersects every vertical straight line Re z=α, a≤α≤b, along an interval. If 1) $$F(z) \in [1,S_{ - \pi \sigma }^{\pi \sigma } \cup K(\alpha + i\pi \sigma ) \cup K(\alpha - i\pi \sigma )], \alpha \in R;$$ 2) 2) $$F( \pm \lambda _n ) = 0, n = 1,2,...,$$ then $$F(z) = A(z)e^{\alpha z} \alpha (z),$$ where $$A(z) \in [1,K], \alpha (z) = \prod\limits_1^\pi {\left( {1 - \frac{{z^2 }}{{\lambda _n^2 }}} \right)}$$ . This result generalizes the theorem of Kaz'min [3]. Three corollaries are also proved, which generalize the theorems ofBoas [1] andPólya [6]. In the theorems of Boas and Pólya, we haveF(n)=0, ?n ε Z. In our case $$F( \pm \lambda _n ) = 0, 0< \lambda _n \uparrow \infty , \sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ .     

The consistencies ofMA,SOCA, OCA andISA withKT(ω2)

In this paper, we prove that ifZFC is consistent, then so are the following theories: $$\begin{gathered} ZFC + MA + KT(\omega _2 ) + 2^{\aleph _0 } = \aleph _2 , \hfill \\ ZFC + SOCA + KT(\omega _2 ), \hfill \\ ZFC + SOCA1 + KT(\omega _2 ), \hfill \\ ZFC + OCA + KT(\omega _2 ), \hfill \\ ZFC + ISA + KT(\omega _2 ), \hfill \\ \end{gathered} $$ whereMA denotes Martin's axiom.KT(ω ₂) the statement:“There exists anω ₂-Kurepa tree”, andSOCA, SOCA1,OCA andISA are axioms introduced in [1].     

The asymptotic behavior of the spectral function for elliptic operators in an unbounded region

We consider elliptic self-adjoint differential operators L of order 2m in a bounded region D? R_n. An asymptotic formula for the function N(λ) = \(N(\lambda ) = \sum\limits_{\lambda _n< \lambda } 1 \) the number of eigenvalues of the operator L less than A. is proved: $$N(\lambda ) = M_0 \lambda ^{n/2m} + o(\lambda ^{n/2m} )$$ whereλ → + ∞ and M₀ is the following constant: $$M_0 = \frac{{V_D }}{{(2\pi )^n \Gamma (1 + n/2m)}}\int_{R_n } {e^{ - L(s)} ds} .$$      

Evaluation of the limits of maximal means

It is proved that the limit $$\mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \tfrac{1}{\Delta }\int_0^\Delta {f(\gamma (t))dt} $$ , wheref: ? → ? is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation γ ∈ [ω1, ω2], coincides with the limit $$\mathop {\lim }\limits_{T \to \infty } \mathop {\sup }\limits_{c \geqslant 0} \varphi _f (k,{\mathbf{ }}T,{\mathbf{ }}c)$$ , where $$\varphi _f = \frac{{(k - 1)\bar I_f (T,c)}}{{1 + (k - 1)\bar \lambda _f (T,c)}},k = \frac{{\omega _2 }}{{\omega _1 }}$$ . Here ¯λf = λf /T, ¯ I_f =If/T, and λf is the Lebesgue measure of the set $$\{ \gamma \in [\gamma _0 ,\gamma _0 + T]:f(\gamma ) \geqslant c\} = A_f ,I_f = \int_{A_f } {f(\gamma )d\gamma } $$ . It is established that this limit always exists for almost-periodic functionsf.     

Eigenvalue asymptotics and a non-linear Schrödinger equation

The equation 1 $$ - \Delta u + qu + f(x,u) = \lambda u, u \in W^{1,2} (\mathbb{R}^N )$$ is considered, whereq is bounded below andq(x)→∞ as |x|→∞. Under appropriate conditions on the perturbation termf(x, u) it is shown that given anyr>0, (*) has an infinite sequence (λ _{n, r} ) _{n ∈ N} of eigenvalues, eachλ _{n, r} being associated with an eigenfunctionu _n,r which satisfies \(\smallint _{R^N } \left| {u_{n,r} } \right|^2 = r^2 \) . Information about the behaviour ofλ _{n, r} for largen is provided. The proofs rely on the compactness of the embedding of a certain weighted Sobolov space in anL ^p space; this is proved in §2.     

On a strengthened extremal property of the Carathéodory-Fejér functions

Для заданной на едини чной окружности огра ниченной функцииω(ξ) рассматр ивается усложненная задача а ппроксимации аналит ическими функциями: $$\mathop {\inf }\limits_{\varphi \in H^\infty } \left[ {\left\| {\omega - \varphi } \right\| + \mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k \left| {\lambda _k } \right|} \right],$$ где ∥·∥ понимается вL ^∞,ε _k ≧0 — заданные чис ла, $$\mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k< + \infty ,\varphi (z) = \mathop \Sigma \limits_{k = 0}^\infty \lambda _k z^k .$$ Доказывается, что при всех достаточно малы хε _k экстремальной в этой задаче будет функция обычного наилучшего приближения (та же, что и приε _k =0,k=0, 1, ...). В частности, при $$\omega (\zeta ) = \frac{{\gamma _0 }}{{\zeta ^n }} + \frac{{\gamma _1 }}{{\zeta ^{n - 1} }} + ... + \frac{{\gamma _{n - 1} }}{\zeta }$$ экстремальной оказы вается дробь Каратео дори—Фейера. Переход к двойственн ой задаче позволяет получить т очные оценки для клас са интегралов типа Коши, выделяемого огранич ениями, наложенными на велич ины коэффициентов ря да Тейлора.     

Optimal Cubature Formulas in Weighted Besov Spaces with A ∞ Weights on Multivariate Domains

For the unit ball $\mathit{UB}_{\tau}^{\alpha}(L_{p,w})$ of the weighted Besov space $B_{\tau}^{\alpha}(L_{p,w})$ with an A _∞ weight w on the domain Ω, which denotes either the unit sphere, or the unit ball, or the standard simplex of the Euclidean space ? ^d , the sharp asymptotic order of the quantity $$\mathop{\inf}_{{\lambda}_1, \ldots, {\lambda}_n \in \mathbb{R}\atop{\xi_1,\ldots, \xi_n \in\varOmega }} \sup_{f\in \mathit{UB}_\tau^\alpha(L_{p,w})} \biggl|\int _{\varOmega } f(x) w(x)\, dx-\sum_{j=1}^n{\lambda}_j f(\xi_j) \biggr|$$ is obtained as n→∞. A similar result is also established on unweighted spherical caps.     

Solving stochastic structural optimization problems by RSM-based stochastic approximation methods — gradient estimation in case of intermediate variables

Reliability-based structural optimization methods use mostly the following basic design criteria: I) Minimum weight (volume or costs) and II) high strength of the structure. Since several parameters of the structure, e.g. material parameters, loads, manufacturing errors, are not given, fixed quantities, but random variables having a certain probability distribution P,stochastic optimization problems result from criteria (I), (II), which can be represented by (1) $$\mathop {\min }\limits_{x \in D} F(x)withF(x): = Ef(\omega ,x).$$ Here,f=f(ω,x) is a function on ? ^r depending on a random element ω, “E” denotes the expectation operator andD is a given closed, convex subset of ? ^r . Stochastic approximation methods are considered for solving (1), where gradient estimators are obtained by means of the response surface methodology (RSM). Moreover, improvements of the RSM-gradient estimator by using “intermediate” or “intervening” variables are examined.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

Duality and equilibrium prices in economics of uncertainty

A random variable (RV) X is given aminimum selling price (S) $$S_U \left( X \right): = \mathop {\sup }\limits_x \left\{ {x + EU\left( {X - x} \right)} \right\}$$ and amaximum buying price (B) $$B_p \left( X \right): = \mathop {\inf }\limits_x \left\{ {x + EP\left( {X - x} \right)} \right\}$$ whereU(·) andP(·) are appropriate functions. These prices are derived from considerations ofstochastic optimization with recourse, and are calledrecourse certainty equivalents (RCE's) of X. Both RCE's compute the “value” of a RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of theCsiszár φ-divergence $$I_\phi \left( {p,q} \right): = \sum\limits_{i = 1}^n {q_i \phi \left( {\frac{{p_i }}{{q_i }}} \right)} $$ a generalized entropy function, measuring the distance between RV's with probability vectors p and q. The RCES _U was studied elsewhere, and applied to production, investment and insurance problems. Here we study the RCEB _P, and apply it to problems ofinventory control (where the attitude towards risk determines the stock levels and order sizes) andoptimal insurance coverage, a problem stated as a game between the insurance company (setting the premiums) and the buyer of insurance, maximizing the RCE of his coverage.     

Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK _x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω _a . As an appli cation, it is proved thatf?L _H ^p (ω _a ,dv _λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α₁,?,α _n and ß = (ß₁, —ß). An interesting question is whether the converse holds.     

Perturbation theorems for holomorphic semigroups

The concept of the gap function is used to give new perturbation results for generators of holomorphic semigroups. In particular, we show that if A is the generator of a holomorphic semigroup on a Banach space and , then every closed linear operator C such that for some and

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ _n=1 ^∞ [µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.     

On minor forms of Desargues and Pappus

Let Π be a projective plane coordinatized by a ternary ring (R, F). In addition to the two operations + and ·, defined bya+b =F(a,1,b and \(a \cdot b = F(a,b,0)\) , a third operation * is defined by \(a * b = F(1,a,b),\forall a,b \in R\) Several minor forms of the propositions of Desargues and Pappus are introduced in Π and their geometrical properties are developed. Several algebraic results are obtained in connection with these minor forms. For example, the first minor form of DesarguesD ₁ is proved to be equivalent to each of the following algebraic identities in every (R, F): (1) $$a \cdot c = c \cdot a \Rightarrow F(a,c,b) = F(c,a,b),$$ (2) $$a \cdot (1 + b) = a + a \cdot b,$$ (3) $$a * b = a + b$$ (4) $$F(x,m,k) = (x \cdot m) * k,\forall a,b,c,k,m,x \in R.$$ Some more algebraic identities are characterized byD ₂ andD ₃.     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

Estimate of a polynomial in generalized exponential functions in the convex hull of two domains

Пусть $$f_n (z) = \exp \{ \lambda _n z\} [1 + \psi _n (z)], n \geqq 1$$ гдеψ _n (z) — регулярны в н екоторой односвязно й областиS, λ _n — нули целой функц ии экспоненциальног о ростаL(λ) с индикатрис ой ростаh(?), причем $$|L\prime (\lambda _n )| > C(\delta )\exp \{ [h(\varphi _n ) - \varepsilon ]|\lambda _n |\} \varphi _n = \arg \lambda _n , \forall \varepsilon > 0$$ . Предположим, что на лю бом компактеK?S $$|\psi _n (z)|< Aq^{|\lambda |_n } , a< q< 1, n \geqq 1$$ гдеA иq зависит только отK. Обозначим через \(\bar D\) со пряженную диаграмму функцииL(λ), через \(\bar D_\alpha \) — смещение. \(\bar D\) на векторα. Рассмотр им множестваD ₁ иD ₂ так ие, чтоD ₁ иD ₂ и их вьшуклая обо лочкаE принадлежатS. Пусть \(\bar D_{\alpha _1 } \subset D_1 , \bar D_{\alpha _2 } \subset D_2 \) Доказывается, что сущ ествует некоторая об ластьG?E такая, что \(\mathop \cup \limits_{\alpha \in [\alpha _1 ,\alpha _2 ]} \bar D_\alpha \subset G\) и дляz∈G верна оценка $$\sum\limits_{v = 1}^n {|a_v f_v (z)|} \leqq B\max (M_1 ,M_2 ), M_j = \mathop {\max }\limits_{t \in \bar D_j } |\sum\limits_{v = 1}^n {a_v f_v (t)} |$$ , где константаB не зав исит от {a _v }.     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.     

On solvability of Dirichlet problem for second order elliptic equation

The paper gives some solvability conditions of the Dirichlet problem for the second order elliptic equation $$ - div(A(x)\nabla u) + (\bar b(x),\nabla u) - div(\bar c(x)u) + d(x)u = f(x) - divF(x),x \in Q,u|_{\partial Q} = u_0 \in L_2 (\partial Q) $$ in bounded domain Q ? R _n (n ≥ 2) with smooth boundary ?Q ∈ C ₁. In particular, it is proved that if the homogeneous problem has only the trivial solution, then for any u ₀ ∈L ₂(?Q) and f, F from the corresponding functional spaces the solution of the non-homogeneous problem exists, from Gushchin’s space $ C_{n - 1} (\bar Q) $ and the following inequality is true: $$ \begin{gathered} \left\| u \right\|_{C_{n - 1} (\bar Q)}^2 + \mathop \smallint \limits_Q r\left| {\nabla u} \right|^2 dx \leqslant \hfill \\ \leqslant C\left( {\left\| {u_0 } \right\|_{L_2 (\partial Q)}^2 + \mathop \smallint \limits_Q r^3 (1 + |\ln r|)^{3/2} f^2 dx + \mathop \smallint \limits_Q r(1 + |\ln r|)^{3/2} |F|^2 dx} \right) \hfill \\ \end{gathered} $$ where r(x) is the distance from a point x ∈ Q to the boundary ?Q and the constant C does not depend on u ₀, f and F.     

Arithmetic properties of the values of a set of functions satisfying a system of differential equations

General results were presented in [2] and [3] concerning arithmetic properties of the values at algebraic points of a class of analytic functions satisfying linear differential equations. In the present note we consider the application of these results to the set of functions $$\begin{gathered} ^f (\alpha _k z) = \sum\nolimits_{n = 0}^\infty {\frac{{ \mu (\mu + 1)... (\mu + n - 1) }}{{\lambda (\lambda + 1)... (\lambda + n - 1)}}} (\alpha _k z)^n (k = 1,2,...,m,) \hfill \\ \lambda \ne 0, - 1, - 2,...), \hfill \\ \end{gathered}$$ where α₁, ..., α_m are algebraic numbers; λ and μ are rational numbers; and the functions satisfy a system of linear differential equations.