On certain estimates for orthogonal polynomials depending on parameters

Рассматривается сис тема ортогональных м ногочленов {P _n (z)} ₀ ^∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {а_{n 0} ^∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦а_n¦<1} ₀ ^∞ ; все многочлены {Р _n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P _n (z)} ₀ ^∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва.     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118

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 \(\mathbb{C}\backslash \mathbb{D}\) .     

The domain of regularity of the limit function of a sequence of analytic functions

Letf (z) be an entire function λ_n(n=0,1,2,...) complex numbers, such that the system f(λ_{n n=0} ^∞ is not complete in the circle ¦z¦n(z) have the form \(\sum\nolimits_{k = 0}^{p_n } {\alpha _{nk} } f(\lambda _k \cdot z)\) . We study the properties of the limit function of the sequence Q_n(z) in the case when $$f(z) = 1 + \sum\nolimits_{n = 1}^\infty {\frac{{z^n }}{{P(1)P(2)...P(n)}}} ,$$ . where P(z) is a polynomial having at least one negative integral root.     

Prime links in some skew-polynomial and skew-laurent rings

Let R be a Noetherian commutative ring and a α₁,…,α_n commuting automorphisms of R. Define T = R[θ₁,…,θ_n;α₁,…,α_n] to be the skew-polynomial ring with θ_ir = α_i(r)θ_i and θ_iθ_j= θ_jθ_i, for all i,j ? (1,…,n) and r ? R, and let S = Rθ₁,θ₁:^-1,…,θ_n:,θ_n;^-1;α₁:,…,α_n] be the corresponding skew-Laurent ring. In this paper we show that S and T satisfy the strong second layer condition and characterize the links between prime ideals in these rings.     

Weak-1 generators for a class of subalgebras of l1

Let α ? 0 and let $\text{D(}\text{α}\text{) = {f(z) = ∑}{}_0{}^{\mathrm{\infty }}\text{α}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{¦ ∑}{}_0{}^{\mathrm{\infty }}\text{(n + 1)}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{¦ a}{}_{\mathrm{n}}\text{¦ < ∞}}$. Then D(α) is a subalgebra of l¹. We discuss the weak-1 generators of D(α). We use some of our techniques to prove that if ? is a weak-1 generator of H^∞ and ∥ ? ∥_∞ ? 1, then the composition operator C_? on the Dirichlet space has dense range.     

On the radius of α-convexity of certain classes of starlike functions

LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σ _n=2/^∞ a _n z ⁿ in the unit discD = {z: |z | < 1} which satisfies inD the condition ?′ (z) ?(z)/z ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$ denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ? (z) =z ^?1 + Σ _n=0/^∞ b _n z ⁿ inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionsz?′(z)/?(z) ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$ In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained.     

On α-close-to-convex functions

LetP(α) denote the class of functionsf analytic in the unit discE, withf(0)=0,f(z)≠0 (0<|z|<1) andf′(z)≠0 inE, satisfying the condition $$\int\limits_{\theta _1 }^{\theta _2 } {\operatorname{Re} } \left\{ {a\left( {1 + \frac{{zf''\left( z \right)}}{{f'\left( z \right)}}} \right) + \left( {1 - a} \right)\frac{{zf'\left( z \right)}}{{f\left( z \right)}}} \right\}d\theta > - \pi $$ whenever 0≤θ₁≤θ₂≤θ₁+2π,z=re^iθ r<1 and α is any positive real number. The functions inP(α) unify the classes of close-to-starlike (α=0) and close-to-convex (α=1) functions. We callf∈P(α) and α-close-to-convex function. In this paper we investigate certain properties of the classP(α).     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables

Let B be the unit ball in C ⁿ and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ _n,α that provide a way to extend a locally univalent function ? ? H(U) to a locally univalent mapping F_α ? H(B), where α ? (0,1]. If ? is normalized univalent, then F_α can be imbedded in a Loewner chain. Also if ? ? S?, then F_α is starlike. We show that if ? belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping F_α satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ _n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C ⁿ (for n ≤ 2.)     

Computation of zeros of the alpha exponential function

This paper deals with the function F(α; z) of complex variable z defined by the expansion \(F\left( {\alpha ;z} \right) = \sum\nolimits_{k = 0}^\infty {\frac{{{z^k}}}{{{{\left( {k!} \right)}^\alpha }}}} \) which is a natural generalization of the exponential function (hence the name). Primary attention is given to finding relations concerning the locations of its zeros for α ∈ (0,1). Note that the function F(α; z) arises in a number of modern problems in quantum mechanics and optics. For α = 1/2, 1/3,..., approximations of F(α; z) are constructed using combinations of degenerate hypergeometric functions ₁ F ₁(a; c; z) and their asymptotic expansions as z → ∞. These approximations to F(α; z) are used to approximate the countable set of complex zeros of this function in explicit form, and the resulting approximations are improved by applying Newton’s high-order accurate iterative method. A detailed numerical study reveals that the trajectories of the zeros under a varying parameter α ∈ (0,1] have a complex structure. For α = 1/2 and 1/3, the first 30 complex zeros of the function are calculated to high accuracy.     

The variety of solutions of the singular generalized Cauchy-Riemann System

We prove that the equation $$2\bar z\partial _{\bar z} \bar w = 0_1 z \in G,$$ in whichB(z) ∈C ^∞(G),B ₀(z)=O(|z})α),α>0,z → 0, and $$b(\varphi ) = \sum\limits_{k = - m_o }^m {b_k e^{ik\varphi } } $$ does not have nontrivial solutions in the classC ^∞(G).     

Proper maps which are Lipschitz α up to the boundary

In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cⁿ into the unit ball in C^N which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is C^k on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than C^N ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bⁿ → B^N that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C^∞ -boundary in Cⁿ, then we can find a map F: D → B^N, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f₁, …, F_N): Bⁿ → B^N that is Lipschitz α and f_N(z) = c(1 - Z₁)^1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bⁿ.     

Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

On lacunary statistical convergence of order α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong N_θ^α(p)-summability. Furthermore, some relations between the spaces N_θ^αS_θ^α are examined.     

Differential Inequalities and Starlikeness

We study some differential inequalities in the unit disc which imply starlikeness: for example if ? (z) = z + Σ^∞ _n=2 a_n (?)zⁿ is analytic in D = {z | |z| < 1} and |z(?′′(z) ? 1)| ≤ 1 ? α, z?D for some α ] [0, 1), then ? is one-to-one on D and ? (D) is a starlike domain with respect to the origin.     

Some inequalities of the V. A. Markov type

Let B be a domain in the complex plane, let p_n(z) and P_n(z) be polynomials of degree n where the zeros of P_n(z) lie in , let(z) be a finite function,(z) 0, z . We consider the problem of estimating from above the functions L[p_n(z)]=(z)p_n(z) – wp_n(z), z , if ¦p_n(z)¦ ¦P_n(z)¦ for zB. Under some very general conditions on B, z, (z), and w we prove the inequality ¦L[p_n(z)]¦ ¦L[P_n(z)]¦.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 431–440, April, 1968.     

The completeness of a functional sequence

Let π_n(u) be a sequence of polynomials with a biorthogonal system, and let {? _n (z)} be functions defined in the singly connected domain D. We consider the problem of the completeness of the system $$A(z,\lambda _n ) = \sum\nolimits_{s = 0}^\infty {P_\mathfrak{s} } (z)\pi _s (\lambda _n ),n = 1,2,...,$$ in the class of functions F(z) having the representation $$F(z) = \sum\nolimits_{k = 0}^\infty d _k P_k (z).$$ .     

Best simultaneous approximation in L∞(μ, X)

Let Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L_∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ₀ is a sub‐σ ‐algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in ?ⁿ . We prove that L_∞(μ, Y) is N ‐simultaneously proximinal in L_∞(μ,X), and that if X is reflexive then L_∞(μ₀, X) is N ‐simultaneously proximinal in L_∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)