A Remez-Type Theorem for Homogeneous Polynomials

Remez-type inequalities provide upper bounds for the uniformnorms of polynomials p on given compact sets K, provided that|p(x)| 1 for every x K\E, where E is a subset of K of smallmeasure. In this paper we prove sharp Remez-type inequalitiesfor homogeneous polynomials on star-like surfaces in R^d. Inparticular, this covers the case of spherical polynomials (whend = 2 we deduce a result of Erdélyi for univariate trigonometricpolynomials).     

Zero-Mean Cosine Polynomials which are Non-Negative for as Long as Possible

For a given integer n, all zero-mean cosine polynomials of orderat most n which are non-negative on [0,(n/(n+1))] are found,and it is shown that this is the longest interval [0,] on whichsuch cosine polynomials exist. Also, the longest interval [0,]on which there is a non-negative zero-mean cosine polynomialwith non-negative coefficients is found. As an immediate consequence of these results, the correspondingproblems of the longest intervals [,] on which there are non-positivecosine polynomials of degree n are solved. For both of these problems, all extremal polynomials are found.Applications of these polynomials to Diophantine approximationare suggested.     

An Elementary Proof of Bailey's Bilinear Generating Function for Jacobi Polynomials and of Its q-Analogue

Some simple ideas are used to give an elementary proof of Bailey'sbilinear generating functionof Jacobi polynomials. The proofpresented here depends only upon certain hypergeometric seriesrepresentations of Jacobi polynomials and the familiar Vandermondesummation theorem; indeed, it can be applied mutatis mutandisto derive analogous bilinear generating functions for the littleq-Jacobi polynomials. This paper concludes by outlining an alternativeproof of one such bilinear q-generating function.     

Extremal Polynomials on Discrete Sets

We study asymptotics for orthogonal polynomials and other extremalpolynomials on infinite discrete sets, typical examples beingthe Meixner polynomials and the Charlier polynomials. Followingideas of Rakhmanov, Dragnev and Saff, weshow that the asymptoticbehaviour is governed by a constrained extremal energy problemfor logarithmic potentials, which can be solved explicitly.We give formulas for the contracted zero distributions, thenth root asymptotics and the asymptotics of the largest zeros.1991 Mathematics Subject Classification: 42C05, 33C25, 31A15.     

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p₁,...,p_m, where each p_j isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors p_j are not unique butthe number m of them is, as is the set of the degrees of thep_j. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced.     

Simultaneous prime specializations of polynomials over finite fields

Recently the author proposed a uniform analogue of the Bateman–Hornconjectures for polynomials with coefficients from a finitefield (that is, for polynomials in F_q[T] rather than Z[T]).Here we use an explicit form of the Chebotarev density theoremover function fields to prove this conjecture in particularranges of the parameters. We give some applications includingthe solution of a problem posed by Hall.     

On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on thenumber of possible homotopy types of the fibres of a Pfaffianmap in terms of the format of its graph. In particular, we showthat if a semi-algebraic set SR^m+n, where R is a real closedfield, is defined by a Boolean formula with s polynomials ofdegree less than d, and : R^m+nRⁿ is the projection on a subspace,then the number of different homotopy types of fibres of doesnot exceed s^2(m+1)n(2^m nd)^O(nm). As applications of our mainresults we prove single exponential bounds on the number ofhomotopy types of semi-algebraic sets defined by fewnomials,and by polynomials with bounded additive complexity. We alsoprove single exponential upper bounds on the radii of ballsguaranteeing local contractibility for semi-algebraic sets definedby polynomials with integer coefficients.     

Interval-polynomial stability theory and its applications in testing the strict positive realness of interval transfer functions

Based on value-set geometry and vector operations in the complexplane, this paper improves some early results on the robustD-stability of an interval polynomial. Almost strong Kharitonov-typeresults for some typical stability regions D are presented.Some connections between the critical vertex polynomials withrespect to these stability regions are established. Explicitupper bounds for the number of critical vertex polynomials associatedwith each stability region are derived. We also present a simpledirect procedure for construction of the critical vertex polynomialswith respect to the left-sector stability region. Illustrativeexamples are given. Using the stability theory of interval polynomials,some strong Kharitonov-type results are obtained for strictpositive realness of interval rational functions.     

A remark on the representation of Zolotarev polynomials

Zolotarev polynomials are the polynomials that have minimaldeviation from zero on [–1, 1] with respect to the norm||xⁿ – x^n–1 + a_n–2 x^n–2 + ... + a₁x+ a_n|| for given and for all a_k . This note complements the paper of F. Pehersforfer [J. LondonMath. Soc. (1) 74 (2006) 143–153] with exact (not asymptotic)construction of the Zolotarev polynomials with respect to thenorm L₁ for || < 1 and with respect to the norm L₂ for || 1 in the form of Bernstein–Szegö orthogonal polynomials.For all in L₁ and L₂ norms, the Zolotarev polynomials satisfyexactly (not asymptotically) the triple recurrence relationof the Chebyshev polynomials.     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

Orthogonal polynomials with a resolvent-type generating function

The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

     

Asymptotic Properties of Collocation Matrix Norms 1: Global Polynomial Approximation

This paper shows that a matrix related to the n-point collocationsolution of mth order ordinary differential equations usingglobal polynomials has maximum norm which tends, as n to themaximum norm of an operator related to the differential equationif the collocation points are chosen as the zeros of certainorthogonal polynomials. The results justify a simple a posterioriestimate for the error in the mth derivative of the correspondingsolution. This result is a consequence of a stronger result that the normof the difference between the matrix mentioned above and anothermatrix associated with the differential equation tends to zeroas n .     

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

We prove a general Borg-type result for reflectionless unitaryCMV operators U associated with orthogonal polynomials on theunit circle. The spectrum of U is assumed to be a connectedarc on the unit circle. This extends a recent result of Simonin connection with a periodic CMV operator with spectrum thewhole unit circle. In the course of deriving the Borg-type result we also use exponentialHerglotz representations of Caratheodory functions to provean infinite sequence of trace formulas connected with the CMVoperator U.     

Algebras of Symmetric Holomorphic Functions on lp

The authors study the algebra of uniformly continuous holomorphicsymmetric functions on the ball of l_p, investigating in particularthe spectrum of such algebras. To do so, they examine the algebraof symmetric polynomials on l_p-spaces, as well as finitely generatedsymmetric algebras of holomorphic functions. Such symmetricpolynomials determine the points in l_p up to a permutation.     

A Class of Pencils of Matrices

A linear machine is one in which the time dependent input yis related to the output z by P(D). z = S(D). y where P andS are polynomials in D = d/dt with constant coefficients. Fornumerical computation it is necessary to replace this relationby a set of simultaneous first order differential equationsand this paper shows how to construct such equations by methodswhich extend the results of Gilder (1961). Attention is restrictedto those sets of equations that are of a special form (see (1))which is characterized by the matrix operating on the dependentvariables. This matrix forms a pencil, being linear in D, andthree theorems are given to show how such matrix pencils maybe constructed from the polynomials. The theorems also statethat any matrix pencil with the required properties can be transformedinto the canonical forms given in the theorems by pre- and post-multiplicationby suitable constant non-singular matrices. Thus the variablesof any set of equations having the required properties are linearcombinations of the variables of the equations given by thetheorems. In the paper it is assumed that the degree of P(D)is greater than that of S(D), as otherwise z would be replacedby z₁+Q(D) . y, where Q is the quotient of S(D)/P(D). Also,as the algebriac manipulations are independent of the natureof the polynomials, D is replaced by an indeterminate x andthe coefficients considered to be from an arbitrary field. Fortechnical reasons we rename y and z, y_o and y_n–_m respectively.     

Homogeneous Orthogonally Additive Polynomials on Banach Lattices

The main result in this paper is a representation theorem forhomogeneous orthogonally additive polynomials on Banach lattices.The representation theorem is used to study the linear spanof the set of zeros of homogeneous real-valued orthogonallyadditive polynomials. It is shown that in certain lattices everyelement can be represented as the sum of two or three zerosor, at least, can be approximated by such sums. It is also indicatedhow these results can be used to study weak topologies inducedby orthogonally additive polynomials on Banach lattices. 2000Mathematics Subject Classification 46G25, 46B42, 47B38.     

Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes

New lower bounds are given for the size of a point set in aDesarguesian projective plane over a finite field that containsat least a prescribed number s of points on every line. Thesebounds are best possible when q is square and s is small comparedwith q. In this case the smallest set is shown to be the unionof disjoint Baer subplanes. The results are based on new resultson the structure of certain lacunary polynomials, which canbe regarded as a generalization of Rédei's results inthe case when the derivative of the polynomial vanishes.     

Wandering Domains in Non-Archimedean Polynomial Dynamics

We extend a recent result on the existence of wandering domainsof polynomial functions defined over the p-adic field C_p toany algebraically closed complete non-archimedean field C_K withresidue characteristic p > 0. We also prove that polynomialswith wandering domains form a dense subset of a certain one-dimensionalfamily of degree p + 1 polynomials in C_K[Z]. 2000 MathematicsSubject Classification 12J25 (primary), 37F99 (secondary).     

A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two-Manifolds

We establish the peak point conjecture for uniform algebrasgenerated by smooth functions on two-manifolds: if A is a uniformalgebra generated by smooth functions on a compact smooth two-manifoldM, such that the maximal ideal space of A is M, and every pointof M is a peak point for A, then A = C(M). We also give an alternativeproof in the case when the algebra A is the uniform closureP(M) of the polynomials on a polynomially convex smooth two-manifoldM lying in a strictly pseudoconvex hypersurface in Cⁿ.     

Symmetric Polynomials on Rearrangement-Invariant Function Spaces

The exact representation of symmetric polynomials on Banachspaces with symmetric basis and also on separable rearrangement-invariantfunction spaces over [0, 1] and [0, ) is given. As a consequenceof this representation it is obtained that, among these spaces,l_2n, L_2n[0, 1], L_2n[0, ) and L_2n[0, )L_2m[0, ) where n, m areboth integers are the only spaces that admit separating polynomials.