Quelques théorèmes sur la structure complexe

We prove the following result: If the function Max (log|ω -f ₁(z)|, ..., log|ω -f _k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ ⁿ ), thenf ₁,...,f _k are analytic functions iff ₁,...,f _k are continuous functions onD(k≥2). We prove also some other results.     

Functions of bounded boundary rotation

Classes of functionsU _k, which generalize starlike functions in the same manner that the classV _k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off _α(z) = ∫ ₀ ^z [f'(t)]^α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV _kfunctions contains the disc of radius 1/k centered at the origin, andV _k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.     

Generalization of the fricke theorem on entire functions of finite index

We prove that, for every sequence (a _k) of complex numbers satisfying the conditions Σ(1/|a _k |) < ∞ and |a _k+1| − |a _k | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/|a _k |) is an entire function of finitel-index.     

On the distribution of the fourier spectrum of Boolean functions

In this paper we obtain a general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1} ^N . Roughly speaking, fixingk∈ℤ₊ and assuming thatf is not essentially determined by a bounded number (depending onk) of variables, we have that . The example of the majority function shows that this result is basically optimal.     

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

Optimal quadrature problem on <Emphasis Type="Italic">n</Emphasis>-information for Hardy-Sobolev classes

For β > 0 and an integer r ≥ 2, denote by [(H)\tilde]_￥,b^r\tilde H_{\infty ,\beta }^r those 2π-periodic, real-valued functions f on ℝ, which are analytic in S _β := {z ∈ ℂ: |Im z| < β} and satisfy the restriction |f ^(r)(z)|≤1, z ∈ S _β . The optimal quadrature formulae about information composed of the values of a function and its kth (k = 1, ..., r − 1) derivatives on free knots for the classes [(H)\tilde]_￥,b^r\tilde H_{\infty ,\beta }^r are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.     

Limits of Teichmüller mappings on trajectories

A classical Teichmüller sequence is a sequence of quasiconformal mapsf _i with complex dilatations of the form , where φ is a quadratic differential and 0≤k _i<1 are numbers such thatk _i→1 asi→∞. This situation occurs in the Teichmüller theory when one moves along a Teichmüller geodesic toward the boundary. The central result is that if τ is a vertical trajectory associated to φ, then there is often, for instance if the sequence is normalized so thatf _i fix 3 points, a subsequence such thatf _i tend either toward a constant or an injective map of τ (Theorem 4.1). If the limit is injective, it is an embedding of τ if τ does not contain points such that τ returns infinitely often to every neighborhood of the point. The main idea is to composef _i locally with a map ϱ_i so that the composed mapf _iϱ_i is conformal and coincides withf _i on τ. Normal family arguments are applied to the sequencef _iϱ_i. Various extensions are presented. The research for this paper has been supported by the project 51749 of the Academy of Finland.     

On the Cohen–Lusk theorem

Let G be a finite group and X be a G-space. For a map f: X → ℝ ^m , the partial coincidence set A(f, k), k ≤ |G|, is the set of points x ∈ X such that there exist k elements g ₁,…, g _k of the group G, for which f(g ₁ x) = ⋅⋅⋅ = f(g _k x) holds. We prove that the partial coincidence set is nonempty for G = ℤ _p ⁿ under some additional assumptions. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 61–67, 2007.     

Properties of generalised juxtapolynomials

GivenF(z),f ₁(z), ..,f _n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ _k ^=1/n a _kf_k(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case. This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this paper.     

On a nontraditional method of approximation

We study the approximation of functions f(z) that are analytic in a neighborhood of zero by finite sums of the form H _n (z) = H _n (h, f, {λ _k }; z) = Σ _k=1 ⁿ λ _k h(λ _k z), where h is a fixed function that is analytic in the unit disk |z| < 1 and the numbers λ _k (which depend on h, f, and n) are calculated by a certain algorithm. An exact value of the radius of the convergence H _n (z) → f(z), n → ∞, and an order-sharp estimate for the rate of this convergence are obtained; an application to numerical analysis is given.     

Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ^* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality. To Reiner Kühnau on his 70th birthday     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.     

A note on absolute summability factors

In this paper, by using an almost increasing and δ-quasi-monotone sequence, a general theorem on φ- | C, α | _k summability factors, which generalizes a result of Bor [3] on φ |C, 1| _k summability factors, has been proved under weaker and more general conditions.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Estimates in the corona theorem and ideals ofH ∞ : A problem of T. Wolff

The main result of the paper is that there exist functionsf ₁,f ₂,f inH ^∞ satisfying the “corona condition”

Integration and Lipschitz functions

The aim of the paper is to prove that every f ∈ L ¹([0,1]) is of the form f = , where j _n,k is the characteristic function of the interval [k- 1 / 2 ⁿ , k / 2 ⁿ ) and Σ _n=0^∞Σ _k=1²ⁿ |a _n,k | is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b _n,k ) _n≧0 ^k=1,...,2n of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).      

Full friendly index sets of Cartesian products of two cycles

Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

Poisson Numbers and Poisson Distributions in Subset Surprisology

Given a family of k + 1 real-valued functions f₀ , ?,f_kf_0 , \ldots ,f_k defined on the set { 1, ?,n}\{ 1, \ldots ,n\} and measuring the intensity of certain signals, we want to investigate whether these functions are T₀ , ?,T_k ,T_0 , \ldots ,T_k , the size a of the collection of numbers j ? { 1, ?,n}j \in \{ 1, \ldots ,n\} whose signals f₀ (j), ?,f_k (j)f_0 (j), \ldots ,f_k (j) exceed the corresponding threshold values T₀ , ?,T_kT_0 , \ldots ,T_k simultaneously for all 0, ?,k0, \ldots ,k is surprisingly large (or small) in comparison to the family of cardinalities

On the 2<Emphasis Type="Italic">m</Emphasis>-th power mean of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with the weight of trigonometric sums

Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a ₀ + a ₁ x + ... + a _k x ^k is a k-degree polynomial with integral coefficients such that (p, a ₀, a ₁, ..., a _k ) = 1, for any integer m, we study the asymptotic property of