Orthogonal Rational Functions and Nested Disks

In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {α_n}^∞_n=0be a sequence in the open unit disk in the complex plane, let

Full-size image

( /|α_k|=−1 whenα_k=0), and let

Full-size image

We consider the following “moment” problem: Given a positive-definite Hermitian inner product ·, · on × , find a non-decreasing functionμon [−π, π] (or a positive Borel measureμon [−π,π)) such that

Full-size image

In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that If this series diverges the solution is always unique.     

Inequalities for Cyclic Functions

The nth cyclic function is defined by

Full-size image

We prove that if k is an integer with 1kn−1, then

Full-size image

holds for all positive real numbers x with the best possible constantsα=1 and β= 2n-k over n.     

Weierstrass'' Theorem in Weighted Sobolev Spaces

We characterize the set of functions which can be approximated by polynomials with the following norm

Full-size image

for a big class of weights w₀, w₁, …, w_k     

z-Approximations

Approximation algorithms for NP-hard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ratio may not be very meaningful—for example, if the ratio of the worst solution to the best solution is at most some constant α, then an approximation algorithm with factor α may in fact yield the worst solution! To overcome this hurdle (among others), several authors have independently suggested the use of a different measure which we call z-approximation. An algorithm is an α z-approximation if it runs in polynomial time and produces a solution whose distance from the optimal one is at most α times the distance between the optimal solution and the worst possible solution. The results known so far about z-approximations are either of the inapproximability type or rather straightforward observations. We design polynomial time algorithms for several fundamental discrete optimization problems; in particular we obtain a z-approximation factor of for the

Full-size image

(TSP) (with no triangle inequality assumption). For the TSP this improves to . We also show that if there is a polynomial time algorithm that for any fixed ε > 0 yields an ε z-approximation then P = NP. We also present z-approximations for severa1 other problems such as

Full-size image

, and

Full-size image

.     

On algebraic K-theory of real algebraic varieties with circle action

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S¹-action so that the quotient Y=X/S¹ is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with ,

Full-size image

is onto, where , denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that

Full-size image

for any real algebraic variety X. As an application we will show that for a compact connected Lie group G .     

A high-order method for the approximate identification of the matrix of a linear differential system

For a linear differential system y′ = Ay with unknown matrix A, we approximate A by a particular linear combination of values of the observed exponential matrix

Full-size image

and show that the L₁ norm of the error matrix is O(hⁿ).     

Functions of class Lip(α, p) and their Taylor mean

The object of this paper is to study the rapidity of convergence of the Taylor mean of the Fourier series of ƒ(x) when ƒ(x) belongs to the class Lip(α, p). We show that it is of Jackson order provided that a suitable integrability condition is imposed upon the function

Full-size image

.     

Boundedness by 3 of the Whitney Interpolation Constant

Let the function fC[0,1] satisfy f( . We prove the estimate

Full-size image

.     

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in L_p[−1,1], 0<p∞, of x^λ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

Full-size image

is finite for λ>0 and .     

Restricted T-Universal Functions

We prove the existence of a function which is holomorphic exactly in the unit disk and has universal translates with respect to a prescribed closed set E∂ and satisfies C^∞(∂ \E). If Q is a subsequence of ₀ with upper density d(Q)=1 then the function can be constructed such that in addition

Full-size image

     

Markov-Type Inequalities for Products of Müntz Polynomials

Let Λ(λ_j)^∞_j=0 be a sequence of distinct real numbers. The span of {x^λ₀, x^λ₁, …, x^λ_n} over is denoted by M_n(Λ)span{x^λ₀, x^λ₁, …, x^λ_n}. Elements of M_n(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λ_j)^∞_j=0andΓ(γ_j)^∞_j=0be increasing sequences of nonnegative real numbers. Let

Full-size image

Then

Full-size image

18(n+m+1)(λ_n+γ_m).In particular ,

Full-size image

Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b](0, ∞).     

Biholomorphically invariant families amongst Carleson class

Given α (0, 1), let T_α be the Carleson class of all meromorphic maps ƒ from the unit disk to the extended complex plane with

Full-size image

where ƒ^# and dm mean the spherical derivative of ƒ and Lebesgue area measure on separately. And, let BIT_α and BIT_α,0 be the biholomorphically invariant families (amongst the Carleson class) consisting of those ƒ T_α with sup and lim_{|w| → 1} ||ƒ ο φ_w||_Tα = 0 respectively, where

. The main purpose of this article is to study BIT_α and BIT_α,0 via the Ahlfors-Shimizu characteristic, canonical factorization and bounded holomorphic maps.     

Construction Techniques for Highly Accurate Quasi-Interpolation Operators

Under mild additional assumptions this paper constructs quasi-interpolants in the form

Full-size image

with approximation order ℓ−1, where_h(x) is a linear combination of translatesψ(x−jh) of a functionψinC^ℓ( ). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

Full-size image

, (0δ1) as 0, and for

Full-size image

as .     

Rate of growth of hypercyclic entire functions

Let

Full-size image

be a weighted shift operator on the space of entire functions. Suppose that |a_a| → ∞ monotonically, a particular such operator being the differentiation operator Df = f′.l It is known that T possesses hypercyclic functions, that is, functions f for which is dense in . In this paper a sharp result on the permissible rates of growth to T-hypercyclic entire functions is obtained, and it is shown that for every permissible growth rate there is a dense subspace of T-hypercyclic functions satisfying this growth condition. These results generalise earlier results by the author and by Armitage on the differentiation operator.     

Best Approximation and Cyclic Variation Diminishing Kernels

We study best uniform approximation of periodic functions from

Full-size image

where the kernelK(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results.     

Prophet inequalities for parallel processes

Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {X_{i, j}, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then

Full-size image

and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes.     

An Improvement of the Constants in a Reverse Integral Inequality

We present an upper bound for the ratio [formula], where f is a positive decreasing function satisfying

Full-size image

for all t (0, a]. Our result sharpens an inequality of L. Nania.     

A generalization of a theorem of Pringsheim

We consider noncommutative continued fractions of the form b₀ + a₁(b₁ + a₂(b₂ + a₃(…)⁻¹ c₃)⁻¹ c₂)⁻¹ c₁, (1) where a_n, b_n and c_n are elements of some Banach algebra B and b_n⁻¹ exists. Such expressions play an important role in the numerical investigation of various problems in theoretical physics and in applied mathematics, but up to now their convergence was not studied in the general case. In this paper we prove a theorem which is an extension of a wellknown theorem of Pringsheim and, in particular, guarantees the convergence of (1) under the following hypotheses:

Full-size image

Full-size image

. As an application, we give a generalization of a theorem of van Vleck. The paper closes with an extensive bibliography.     

Interpretation of the Lavrentiev phenomenon by relaxation

We consider functionals of the calculus of variations of the form F(u)= ∝₀¹ f(x, u, u′) dx defined for u ε W^1,∞(0, 1), and we show that the relaxed functional with respect to weak W^1,1(0, 1) convergence can be written as

Full-size image

, where the additional term L(u), called the Lavrentiev term, is explicitly identified in terms of F.