An Entire Holomorphic Function Associated to an Entire Harmonic Function

Let h be a harmonic function on ^N. Then there exists a holomorphic function f on such that f(t)=h(t, 0, …, 0) for all real t. Precise inequalities relating the growth rate of f to that of h are proved. These results are applied to deduce uniqueness theorems for harmonic functions of sufficiently slow growth that vanish at certain lattice points. Another application concerns the rate at which a harmonic function of finite order can decay along a ray.     

On the Denseness of Rational Systems

This note characterizes the denseness of rational systems

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in C[−1, 1], where the nonreal poles in {a_k}^∞_k=1 \[−1, 1] are paired by complex conjugation. This extends an Achiezer's result.     

Approximation in Lp( ) from Spaces Spanned by the Perturbed Integer Translates of a Radial Function

The problem of approximating smooth L_p-functions from spaces spanned by the integer translates of a radially symmetric function φ is very well understood. In case the points of translation, Ξ, are scattered throughout ^d, the approximation problem is only well understood in the “stationary” setting. In this work, we provide lower bounds on the obtainable approximation orders in the “non-stationary” setting under the assumption that Ξ is a small perturbation of ^d. The functions which we can approximate belong to certain Besov spaces. Our results, which are similar in many respects to the known results for the case Ξ= ^d, apply specifically to the examples of the Gauss kernel and the generalized multiquadric.     

Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F:  →  alg_f be a functor, where alg_f is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra A_aut(F) such that F factorizes through a functor :  →  coalg_f(A_aut(F)), where coalg_f(A_aut(F)) is the category of finite-dimensional measured A_aut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category coalg_f(A) and the forgetful functor ω: coalg_f(A) →  alg_f: we have A  A_aut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz.     

Universal Functions on Complex General Linear Groups

In 1929, Birkhoff proved the existence of an entire function F on with the property that for any entire function f there exists a sequence {a_k} of complex numbers such that {F(ζ+a_k)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoff's theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group *. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n,  ) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n,  ) be the general linear group of degree n over . We denote by (G) the set of all holomorphic functions on G. Similarly, we define ( ). Let K be the (G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n,  ) with f(0)(f(K)), where (f(K)) is the ( )-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any KB(G), for any function f holomorphic in some neighbourhood of K, and for any >0, there exists CG with max_ZK |F(CZ)−f(Z)|<.     

Boundedness of generalized higher commutators of Marcinkiewicz integrals

Let (b) ＝ (b1,…,bm) be a finite family of locally integrable functions. Then,we introduce generalized higher commutator of Marcinkiwicz integral as follows:μ(b)Ω=(∫∞o|F(b)Ω,t(f)(x)|2et/t)1/2,whereF(b)Ω(f)(x)＝1/t∫|x-y|≤tΩ(x-y)/|x-y|n-1m∏j=1(bj(x)-bj(y))f(y)dy.When bj ∈(A)βj, 1≤j≤m, 0＜βj＜1,m∑j=1βj =β＜n, and Ω is homogeneous of degree zero and satisfies the cancelation condition, we prove that μ(b)Ω is bounded from Lp(Rn)to Ls(Rn), where 1 ＜ p ＜ n/β and 1/s = 1/p -β/n. Moreover, if Ω also satisfies some Lq-Dini condition, then μ(b)Ω is bounded from Lp(Rn) to (F)β,∞p(Rn) and on certain Hardy spaces. The article extends some known results.     

Markov-type inequalities on certain irrational arcs and domains

Let denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set KR^d set Then the Markov factors on K are defined by (Here, as usual, S^d-1 stands for the Euclidean unit sphere in R^d.) Furthermore, given a smooth curve ΓR^d, we denote by D_TP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by Let . We prove that for every irrational number α>0 there are constants A,B>1 depending only on α such that for every sufficiently large n.Our second result presents some new bounds for M_n(Ω_α), where (d=2,α>1). We show that for every α>1 there exists a constant c>0 depending only on α such that M_n(Ω_α)n^clogn.     

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}_j=1ⁿ be n distinct points and let L_n[·] denote the corresponding Lagrange Interpolation operator. Let W : →[0,∞). What conditions on the array {x_jn}_{1jn, n1} ensure the existence of p>0 such

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for every continuous f : → with suitably restricted growth, and some “weighting factor” φ^b? We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set.     

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space (k) shares certain properties with the classical Bergman space L²_α of the unit disk. For example, the weighted Bergman kernels k^β_w(z)=(1−wz)^−β, 1β2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H² (k)L²_a, where the inclusion maps are contractive, and M_ζ, the operator of multiplication with the identity function ζ, defines a contraction operator on (k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace of (k) is a Bergman-type kernel as well. For the weighted Bergman kernels k^β this result even holds for all _ζ-invariant subspace of index 1, i.e., whenever the dimension of /ζ is one. Second, if is any multiplier invariant subspace of (k), and if we set *= z , then M_ζ is unitarily equivalent to M_ζ acting on a space of *-valued analytic functions with an operator-valued reproducing kernel of the type

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where V is a contractive analytic function V :  → ( ,  *), for some auxiliary Hilbert space . Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of (k^β), 1β2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.     

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|_n|²dσ}_n0, denoted by Lim(σ). Here {_n}_n0 are orthogonal polynomials in L²(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m( )=1) Lebesgue measure on . The second subset Mar( ) consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar( ) iff there are >0 and l>0 such that sup{|a_n+j|:0jl}>, n=0,1,2,…,{a_n} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ with {m}Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σMar( )). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zⁿ=λ, λ , n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zⁿ, n=1,2,…, of a closed arc J (including J= ) with removed open concentric arc J₀ (including J₀=). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).     

On Identity Bases of Epigroup Varieties

Let C _n and C _n be the varieties of all completely regular and of all completely simple semigroups, respectively, whose idempotent generated subsemigroups are periodic with period n. We use Ol'shanski 's theory of geometric group presentations to show that for large odd n these varieties (and similarly defined varieties of epigroups) do not have finitely axiomatizable equational theories.     

Strong bias of group generators: an obstacle to the “product replacement algorithm”

Let G be a finite group. Efficient generation of nearly uniformly distributed random elements in G, starting from a given set of generators of G, is a central problem in computational group theory. In this paper we demonstrate a weakness in the popular “product replacement algorithm,” widely used for this purpose. The main results are the following. Let be the set of generating k-tuples of elements of G. Consider the distribution of the first components of the k-tuples in induced by the uniform distribution over  . We show that there exist infinite sequences of groups G such that this distribution is very far from uniform in two different senses: (1) its variation distance from uniform is >1−ε; and (2) there exists a short word (of length (loglog|G|)^O(k)) which separates the two distributions with probability 1−ε. The class of groups we analyze is direct powers of alternating groups. The methods used include statistical analysis of permutation groups, the theory of random walks, the AKS sorting network, and a randomized simulation of monotone Boolean operations by group operations, inspired by Barrington's work on bounded-width branching programs. The problem is motivated by the product replacement algorithm which was introduced in [Comm. Algebra 23 (1995) 4931–4948] and is widely used. Our results show that for certain groups the probability distribution obtained by the product replacement algorithm has a bias which can be detected by a short straight line program.     

Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

Best Approximation by Normal and Conormal Sets

The aim of the present paper is to develop a theory of best approximation by elements of so-called normal sets and their complements—conormal sets—in the non-negative orthant ^I₊ of a finite-dimensional coordinate space ^I endowed with the max-norm. A normal (respectively, conormal) set arises as the set of all solutions of a system of inequalities f_α(x)0 (αA), x ^I₊ (respectively, f_α(x)0 (αA), x ^I₊), where f_α is an increasing function and A is an arbitrary set of indices. We consider these sets as analogues (in a certain sense) of convex sets, and we use the so-called min-type functions as analogues of linear functions. We show that many results on best approximation by convex and reverse convex sets and corresponding separation theory (but not all of them) have analogues in the case under consideration. At the same time there are no convex analogues for many results related to best approximation by normal sets.     

The Linear Part of a Discontinuously Acting Euclidean Semigroup

Let ⁿ be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of ⁿ. We say that a semigroup S acts discontinuously on ⁿ if the subset {s  S:sK ∩ K ≠ } is finite for any compact set K of ⁿ. The main results of this work areTheorem.If S is a Euclidean semigroup which acts discontinuously on ⁿ, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group.Corollary.Let S be a Euclidean semigroup acting discontinuously on ⁿ; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n).These results are the first step in the proof of the followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group.     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.