Upper bounds for the autocorrelation coefficients of the Rudin-Shapiro polynomials

Givena _m to be them ^th correlation coefficient of the Rudin-Shapiro polynomials of degrees 2 ⁿ ? 1, ¦a _m¦ ≤ C(2 ⁿ )^3/4 and there existsk ≠ 0 such that ¦a _k¦ >D(2 ⁿ )^0.73 (C andD are universal constants). Here we show that the 0.73 is optimal in the upper bound case.     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

Analytic Extension of Smooth Functions

Let F be a closed proper subset of ?ⁿ and let ?* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ? ∈ ?* (?ⁿ) there is ${\widetilde f} \in {\cal E}_{*}(\rm R ^{n})$ which is real analytic on ?ⁿF and such that ?^a ? ¦ _F = ?^a ? ¦ _F for any a ∈ ?₀ ⁿ. For bounded ultradifferentiable functions ? we can obtain ${\widetilde f}$ by means of a continuous linear operator.     

Estimates in the Carleson corona theorem,ideals of the algebra H∞, a problem of S.-Nagy

Let E₁, E₂, be Hilbert spaces, H^∞(E₁,E₂) be the space of functions, bounded and analytic in the disk D, with values in the space of bounded linear operators from E₁ to E₂. Estimates are investigated for a solution of the problem of S.-Nagy of finding a left inverse element for a function F, FεH^∞(E₁,E₂). For dim E₁=1 this problem is a generalization of the corona problem. Let C_n(δ)= sup_¦G¦_∞∶FεH^∞(E₁,E₂),dim E₁=n, ¦F¦_∞?1, ¦F(z)a¦₂?δ¦a¦₂(zεD,aεE₁);Gε H^∞(E₂,E₁) is a function of minimal norm for which . Then where a_n, C_n are constants depending only on n. The behavior of the function C₁ as δ→1 is described. Other results are obtained also.     

Extremal functional interpolation in the mean with least value of thenth derivative for large averaging intervals

The smallest numberA<∞ is found such that for any sequenceY={y _k ,k ∈ ?} with ¦Δ ⁿ y _k ¦≤1 there exists au(t), ¦u(t)¦ ≤ A, for which the equationy ⁿ (t)=u(t) (?∞<t<∞) has a solution satisfying the conditions $$y_k = \frac{1}{h}\int_{ - h/2}^{h/2} {y(k + 1){\mathbf{ }}dt} ,{\mathbf{ }}where{\mathbf{ }}k{\mathbf{ }} \in {\mathbf{ }}\mathbb{Z},{\mathbf{ }}1{\mathbf{ }}< {\mathbf{ }}h{\mathbf{ }}< {\mathbf{ }}2.$$ , wherek ∈ ?, 1<h<2. A similar problem is treated inL _p (?∞, ∞). It is shown that forh=2m (m a natural number) no such finiteA exists.     

Extremal property of some surfaces in n-dimensional Euclidean space

A surface Γ=(f ₁(X₁,..., x_m),...,f _n(x₁,..., x_m)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a _i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa ₁, ...,a _n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf ₁, ...,f _n, the extremality of a class of surfaces in n-dimensional Euclidean space.     

Некоторые точные инт егральные оценки про изводных рациональных и алгеб раических функций. Пр иложения

Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results:

1. IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝_E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure;
2. Iff(x ₁,x ₂, ...,x _m) is a real valued piecewise algebraic function of order (n, k) on the unit ballD?R ^m (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝_D¦gradf¦dx≦2π ^m/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞;
3. LetE=Π={z∶¦z¦=1}, and letc _m(R) be the mth Laurent coefficient ofR onΠ,C _m(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦C _m(n) ≦ min {n/¦m¦, 1}.

     

Integral Domains with Highly Nonunique Factorization

For a positive integer n, an atomic integral domain R is defined to be completely non- n- factorial if for any n atoms a₁…, a_n, the product a₁ … a _n has as highly nonunique a factorization into atoms as possible in that given any n ? 1 atoms b₁,…, b_{nt - 1}, b₁ … b _n? 1¦a₁ … a _n. We show that R is completely non-n-factorial for some n ≥ 2 if and only if (R, M) is a quasilocal domain with [M: M] a DVR having M as its maximal ideal.     

On optimal extrapolation and interpolation of fuzzy analytic functions

Для класса ? аналитич еских в единичном кру ге функций, ограниченны х по модулю единицей, погрешност ью наилучшего прибли жения в точкеz ₀ по значениям в точкахz ₁,..., z_n, заданным с погрешнос тьюδ, называется вели чинаr(z ₀, z₁ z..., z_n, α)=inf sup sup ¦f(z₀)-S(f₁, ...f_n)¦, где нижняя грань бере тся по всевозможным ф ункциям S: Сⁿ→С. ДляE～((?1,1) иz ₀∈ ∈(-1,1)Е рассматривается задача о нахождении п орядка информативности мно жестваЕ, т.е. минимальногоп, на котором достигается нижняя грань в равенстве $$R(z_0 ,\delta ,E) = \mathop {\inf }\limits_n {\text{ }}\mathop {\inf }\limits_{z_1 , \ldots ,z_n \in E} {\text{ }}r(z_0 ,z_1 , \ldots ,z_n ,\delta ).$$ Кроме того, приδ, близ ких к 1, решена задача о нахождении величины $$r_n (\delta ,E) = \mathop {\inf }\limits_{z_1 , \ldots ,z_n \in Ez_0 \in E} \sup r(z_0 ,z_1 , \ldots ,z_n ,\delta )$$ и найдены узлы, на кото рых достигается нижн яя грань.     

Permutation Polynomials Modulo 2w

We give an exact characterization of permutation polynomials modulo n=2^w, w≥2: a polynomial P(x)=a₀+a₁x +···+a_dx^d with integral coefficients is a permutation polynomial modulo n if and only if a₁ is odd, (a₂+a₄+a₆+···) is even, and (a₃+a₅+a₇+···) is even. We also characterize polynomials defining latin squares modulo n=2^w, but prove that polynomial multipermutations (that is, a pair of polynomials defining a pair of orthogonal latin squares) modulo n=2^wdo not exist.     

Uniform and mean approximation by certain weighted polynomials,with applications

LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa _n =a _n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP _n(x) of degree at mostn, the sup norm ofP _n(x)W(a _n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p _n (x)W(a _n x)} ₁ ^∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p _n (x)W(a _n x)} ₁ ^∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL _p analogue for 0W(x)=exp(?|x| ^α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| ^p exp(?|x| ^α ),α > 0,p > ?1.     

Coefficient inequalities for a subclass of starlike functions

A function f(z) = z ? ∑^∞_{n = 2}a_nzⁿ, a_n ? 0, analytic and univalent in the unit disk, is said to be in the family $\text{T}{}^1\text{(a, b)}$, a real and b ? 0, if $\text{¦(}\text{zf′}\text{f}\text{) ? a¦ ? b}$ for all z in the unit disk. A complete characterization is found for $\text{T}{}^1\text{(a, b)}$ when a ? 1. Also, sharp coefficient bounds are determined for certain subclasses of $\text{T}{}^1\text{(a, b)}$ when a < 1; however, examples are given to show that these bounds do not remain valid for the whole family.     

Базисы из экспонент в пространствахL p

Let the rootsλ _n of an entire functionL(z) be separated and lie in some horizontal strip ¦Im z¦ ≦h, and suppose that $$0< c \leqq |L(z)|(1 + |z|)^{ - b} \exp ( - a|\operatorname{Im} z|) \leqq C< \infty$$ for ¦Imz¦≧H>h. If 1<p<2 and - 1/pq (1/q+1/p=1), then the system {exp (iλ _n x)} _n=0 ^∞ constitutes a basis нn the spaceL ^p (-a,a). In the caseb=1/q orb=?1/p the theorem fails, Equivalence of the following two statements is also proved:

1. {exp (iλ _n x)} _n=0 ^∞ is an extendable convergence system inL ^p from the interval (-a, a).
2. {exp (iλ _n x)} _n=0 ^∞ is a continuable basis inL ^p (-a,a).

     

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.     

The mean square of Dirichlet series associated with automorphic forms

Letf be a non-holomorphic automorphic form of real weight and eigenvalue λ=1/4?ρ ², ?ρ≥0, which is defined with respect to a Fuchsian group of the first kind. Assume that ∞ is a cusp of this group and denote bya _∞,n,a _∞,n ,n ∈ ?, the Fourier coefficients off at ∞. Following Hecke and Maas we prove that under suitable assumptions the associated Dirichlet seriesL ⁺ (f, s) = ∑ _{n > 0} a _∞,n (n + μ_221E;)^?s andL ^? (f, s) = ∑ _{n < 0} a _∞,n |n + μ_221E;|^?s have meromorphic continuation in the entire complex plane and statisfy a certain functional equation (μ_∞ denotes the cusp parameter of the cusp ∞). We are interested in mean square estimates of these functions. Iff is not a cusp form we prove $$\int_0^T {|L^ \pm (f,\Re _\rho + it)|^2 dt = T(\log T)^a (B^ \pm + o(1)),}$$ wherea is either 1, 2 or 4, andB ^± is a constant. A similar result is true iff is a cusp form. In case of a congruence group the termo(1) can be replaced byO ((logT)^?1).     

On a transformation of power series

Изучаются ряды Риман а, рассматривавшиеся ранее в работах [1] и [2]. Пустьa _n (n=1, 2,…) — последов ательность комплекс ных чисел иr _n =a _n +a _2n +…. Предполо жим, чтоΣ¦a _n ¦<∞. Тогда выпо лняются неравенства $$\begin{array}{*{20}c} {\sum\limits_n {\left| {r_n } \right| \leqq } \sum\limits_n {\left| {a_n } \right|} d(n),} & {\sum\limits_n {\left| {a_n } \right|} } \\ \end{array} \leqq \sum\limits_n {\left| {r_n } \right|2^{\omega (n)} ,} $$ гдеd(n) иω(n) — соответств енно число делителей и число простых делителейn. Е сли $$\begin{array}{*{20}c} {F(z) = \sum\limits_n {a_n z^n ,} } & {p_n (z) = \sum\limits_{s|n} {\mu \left( {\frac{n}{s}} \right)z^s ,} } \\ \end{array} $$ то \(F(z) = \sum\limits_n {r_n p_n (z)} \) для ¦z¦<1. В статье с одержатся некоторые результаты о сходимо сти рядов РиманаΣt _n p _n (z) на окружно сти ¦z¦=1. Например, если числаt _n неотрицатель ны, монотонно убывают и \(\sum\limits_n {t_n< \infty } \) , то ряд равн омерно сходится для ¦z¦=1. Сформулированы неко торые новые задачи.     

On separable Lindenstrauss spaces

Isometric embeddings from l_∞ⁿin l_∞^{n + 1} can be described by a_i,n, i ? n, with $\text{∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{¦ a}{}_{\mathrm{i,n}}\text{¦ ? 1}$, such that e_i,n = e_{i,n + 1} + a_i,ne_{n + 1,n + 1}; i = 1,…, n; holds, where e_i,nand e_{i,n + 1} are the elements of the canonical unit vector bases of l_∞ⁿand l_∞^{n + 1}, respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a_i,n, i = 1,…, n, and the Banach space a_i,n, E_n ? E_{n + 1}, E_n ? l_∞ⁿ, where A determines the embeddings of the E_n. The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c₀and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C(K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A(S) where S is a Choquet simplex with dense extreme points.     

Bilinear and quadratic variants on the Littlewood-Offord problem

If f(x ₁, …, x _n ) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a ₁, …,a _n , what is the maximum number of sums of the form ±a ₁ ± a ₂ ± … ± a _n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n ^?1 are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n ^?1/2. In both cases the results are nearly tight.     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.