Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ² the set of convex functions f ∈ ℂ[−,1]. Also, let E _n (f) and E _n ⁽²⁾ (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ² by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E _n ⁽²⁾ (f), and Lorentz and Zeller proved that the inverse inequality E _n ⁽²⁾ (f) ≦ cE _n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ². In this paper we prove, for every α > 0 and function f ∈ Δ², that

Exact asymptotic behaviour of the codimensions of some P.I. algebras

Letc _n (A) denote the codimensions of a P.I. algebraA, and assumec _n (A) has a polynomial growth: . Then, necessarily,q∈ℚ [D3]. If 1∈A, we show that , wheree=2.71…. In the non-unitary case, for any 0<q∈ℚ, we constructA, with a suitablek, such that . In memory of S. A. Amitsur, our teacher and friend Partially supported by Grant MM404/94 of Ministry of Education and Science, Bulgaria and by a Bulgarian-American Grant of NSF. Partially supported by NSF grant DMS-9101488.     

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

Strictly Hermitian positive definite functions

LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix

Sharp estimates for maximal operators associated to the wave equation

The wave equation, ∂ _tt u=Δu, in ℝ ⁿ⁺¹, considered with initial data u(x,0)=f∈H ^s (ℝ ⁿ ) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H ^s (ℝ ⁿ ) to L ^q (ℝ ⁿ ).     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums

Boundary constructions of petals at the Wolff point in the parabolic case

In the same spirit of the classical Leau-Fatou flower theorem, we prove the existence of a petal, with vertex at the Wolff point, for a holomorphic self-map f of the open unit disc Δ ⊂ ℂ of parabolic type. The result is obtained in the framework of two interesting dynamical situations which require different kinds of regularity of f at the Wolff point τ: f of non-automorphism type and or f injective of automorphism type, f∈C ^3+ɛ(τ) and . Partially supported by PRIN Proprietà geometriche delle varietà reali e complesse. Partially supported by GNSAGA of the Istituto Nazionale di Alta Matematica, Rome.     

Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

Let and τ denote the invariant gradient and invariant measure on the unit ball B of ℂⁿ, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C²(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H _ϕ(B>) if and only if

A formula for the Bloch norm of a C 1-function on the unit ball of ℂ n

For a C ¹-function f on the unit ball ⊂ ℂ ⁿ we define the Bloch norm by , where is the invariant derivative of f, and then show that . Supported by MNZŽS Serbia, Project No. 144010.     

On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let

On the equivalence of some conditions for weighted Hardy spaces

Let G ∈ H _σ ^p (ℂ₊), where H _σ ^p (ℂ₊) is the class of functions analytic in the half plane ℂ₊ = {z: Re z > 0} and such that

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

Almost holomorphic extensions of ultradifferentiable functions

For a smooth functionf on ℝ ⁿ , we construct an extensionF to ℂ ⁿ with vanishing to a high order on ℝ ⁿ and give precise estimates of how the degree of smothness is reflected in the degree of vanishing. This analysis is used to define the operator on (n,n−1) forms with singularities on ℝ ⁿ .     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

Nonisospectral flows on semiinfinite unitary block Jacobi matrices

It is proved that if the spectrum and the spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix J(t) vary appropriately, then the corresponding operator J(t) satisfies the generalized Lax equation , where Φ(gl, t) is a polynomial in λ and with t-dependent coefficients and is a skew-symmetric matrix. The operator J(t) is analyzed in the space ℂ ⊕ ℂ² ⊕ ℂ² ⊕ …. It is mapped into the unitary operator of multiplication L(t) in the isomorphic space , where . This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation. A procedure that allows one to solve the corresponding Cauchy problem by the inverse-spectral-problem method is presented. The article contains examples of block difference-differential lattices and the corresponding flows that are analogs of the Toda and the van Moerbeke lattices (from the self-adjoint case on ℝ) and some notes about the application of this technique to the Schur flow (the unitary case on and the OPUC theory). Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 521–544, April, 2008.