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.     

On Some Extremal Properties of Lagrange Interpolatory Polynomials

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show that

Full-size image

, where _n is a set of the linear operators with finite rank n+1 defined on −1,1], and where _n+1 denotes the set of polynomials p=∑_i=0ⁿ⁺¹a_ixⁱ of degreen+1 such that a_n+11. The infimum is achieved for Lagrange interpolatory polynomial for nodes .     

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.     

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     

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, ∞).     

A generalization of Everett's result on the Collatz 3x + 1 problem

It is a well-known conjecture that given m ε N, the set of natural numbers, the sequence {m_n}^∞_n−0, defined by the iterative formula m₀ = m,

Full-size image

has some iterate m_j = 1. It is shown in this paper that for any k ε N, “almost every” natural number m greater than unity has k iterates less than m.     

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).     

Oblique Projections and Abstract Splines

Given a closed subspace of a Hilbert space and a bounded linear operator AL( ) which is positive, consider the set of all A-self-adjoint projections onto :

Full-size image

In addition, if ₁ is another Hilbert space, T: → ₁ is a bounded linear operator such that T^*T=A and ξ , consider the set of (T, ) spline interpolants to ξ:

Full-size image

A strong relationship exists between (A, ) and sp(T, ,ξ). In fact, (A, ) is not empty if and only if s p(T, ,ξ) is not empty for every ξ . In this case, for any ξ it holds

Full-size image

and for any ξ , the unique vector of s p(T, ,ξ) with minimal norm is (1−P_A, )ξ, where P_A, is a distinguished element of (A, ). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.     

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.     

Asymptotics for Lp extremal polynomials on the unit circle

Let p > 1, and dμ a positive finite Borel measure on the unit circle Γ: = {z ε C: ¦z¦ = 1}. Define the monic polynomial φ_{n, p}(z)=zⁿ+…εP_n >(the set of polynomials of degree at most n) satisfying

Full-size image

. Under certain conditions on dμ, the asymptotics of φ_{n, p}(z) for z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φ_{n, p} are also discussed (cf. Theorems 3.1 and 3.2).     

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 .     

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 .     

Boundedness by 3 of the Whitney Interpolation Constant

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

Full-size image

.     

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.     

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.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

Full-size image

where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

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

.     

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

.     

On the Denseness of Rational Systems

This note characterizes the denseness of rational systems

Full-size image

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.