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1.
We consider orthogonal polynomials , where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight WN which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit and link this limit to an external field for an equilibrium problem in logarithmic potential theory.  相似文献   

2.
We consider orthogonal polynomials on the real line with respect to a weight and in particular the asymptotic behaviour of the coefficients an,N and bn,N in the three-term recurrence xπn,N(x)=πn+1,N(x)+bn,Nπn,N(x)+an,Nπn−1,N(x). For one-cut regular V we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients an,n and bn,n have asymptotic expansions as n in powers of 1/n2 and powers of 1/n, respectively.  相似文献   

3.
Letk>1 and let be non-zero algebraic numbers contained in the field . It is shown that for almost all, in the sense of density integer vectorsn 1,...,n k the polynomial becomes irreducible over on dividing by the product of all factorsx–, where is a root of unity.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday  相似文献   

4.
Résumé Pour des variables matriciellesn x n, nous démontrons la réductibilité en degré de certains produits et de leurs traces, afin de généraliser au delà des dimensions 2 ou 3 quelques résultats employés parRivlin, Smith etSpencer, pour extraire une base d'intégrité orthogonale des tenseurs symétriques. En particulier, pourn3, nous démontrons la réductibilité en degré de tout produit de 2 n –1 matrices et de la trace de tel produit.

The work described in this paper was carried out under Contract No. 562 (40).  相似文献   

5.
We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of the integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for the integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for the integer Chebyshev constant. Moreover, all the bounds mentioned can be found numerically by using various extremal point techniques, such as the weighted Leja points algorithm. Applying our results in the classical case of the segment [0, 1], we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials. Research supported in part by the National Security Agency under Grant No. MDA904-03-1-0081.  相似文献   

6.
Factoring polynomials with rational coefficients   总被引:61,自引:0,他引:61  
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7.
8.
We obtain other refinements of the inequalities of S. N. Bernstein and M. Riesz for polynomials. The methods of proof use the theory of boundpreserving convolution operators in the unit disk and interpolation formulas.  相似文献   

9.
Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.  相似文献   

10.
11.
Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 150–153, January, 1995.  相似文献   

12.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 12, pp.1669–1680, December, 1989.  相似文献   

13.
In this paper we investigate general properties of the coefficients in the recurrence relation satisfied by multiple orthogonal polynomials. The results include as particular cases Angelesco and Nikishin systems.  相似文献   

14.
We establish sufficient conditions of the reducibility of the linear system of difference equationsx(t+1)=Ax(t) + P(t) x(t) with an almost periodic matrixP(t) to a system with a constant matrix.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1661–1667, December, 1993.  相似文献   

15.
We prove that an algebraic number α is a root of a polynomial with positive rational coefficients if and only if none of its conjugates is a nonnegative real number. This settles a recent conjecture of Kuba.  相似文献   

16.
17.
We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2mn, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1].  相似文献   

18.
This paper describes the main properties of analytic common monic left multiples of a system of monic operator polynomials whose coefficients depend analytically on .This paper was written while the first author was a visiting professor of the Vrije Universiteit at Amsterdam  相似文献   

19.
We prove that the roots of a definable C curve of monic hyperbolic polynomials admit a definable C parameterization, where ‘definable’ refers to any fixed o-minimal structure on (ℝ,+, ·). Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of C p (for p ∈ ℕ) arrangements of the roots in both the definable and the non-definable case. These conditions are sharp in the definable and, under an additional assumption, also in the non-definable case. In particular, we obtain a simple proof of Bronshtein’s theorem in the definable setting. We prove that the roots of definable C curves of complex polynomials can be desingularized by means of local power substitutions t ↦ ±t N . For a definable continuous curve of complex polynomials we show that any continuous choice of roots is actually locally absolutely continuous.  相似文献   

20.
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