On the Zeros of Orthogonal Polynomials: The Elliptic Case

Constructive Approximation - Let E = [–1, α] \cup [β, 1], –1 &;lt; α &;lt; β &;lt; 1, and let (pn) be orthogonal on E with respect to the weight function...     

Perturbed recurrence relations II the general case

In this paper, we give some new explicit relations between two families of polynomials defined by recurrence relations of all order. These relations allow us to analyze, even in the Sobolev case, how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation and the order are perturbed. In a paper we have already given a method which allows us to study the polynomials defined by a three-term recurrence relation. Also here some generalizations are given.     

一类非线性复微分方程的超越亚纯解北大核心CSCD

本文主要研究非线性复微分方程f^(4)+a(z)ff^((k))=p_(1)(z)e^(α_(1)(z))+p_(2)(z)e^(α_(2)(z))的超越亚纯解,其中a,p_(1),p_(2)是非零的有理函数,α_(1),α_(2)是非常数的多项式.进一步地,考虑当亚纯解存在时,α_(1),α_(2),p_(1)和p_(2)所满足的条件.另外,还讨论了非线性复微分方程f^(3)+a(z)f’=p_(1)(z)e^(ν(z))+p_(2)(z)e^(-ν(z))的亚纯解的存在性,其中a,p_(1),p_(2)是非零的有理函数,ν是非常数的多项式.所得的结果直接推广了一些已知的结果.     

Spectrum is periodic for n-intervals

In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set Ω is periodic, with the period an integral multiple of the measure of Ω. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.     

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Boros and Moll via

     

Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations

This paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator: its self-adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations. The sufficient conditions given in both cases improve and generalize previously known results. It turns out that, not only self-adjointness helps to study limit points of eigenvalues of truncated operators, but the analysis of such limit points is a key help to prove self-adjointness. Several examples show the advantages of these new results compared with previous ones. Besides, an application to the theory of continued fractions is pointed out.     

Some Indeterminate Moment Problems and Freud-Like Weights

We study two indeterminate Hamburger moment problems and the corresponding orthogonal polynomials. The coefficients in their recurrence relations are of exponential growth or are polynomials of degree 2. The entire functions in the Nevanlinna parametrization are found. The orthogonal polynomials with polynomial recurrence coefficients resemble the Freud polynomials with a = 1/2 . Inequalities are given for the largest zero and the asymptotic behavior of the largest zero is established. April 24, 1996. Date revised: March 3, 1997.     

A companion matrix approach to the study of zeros and critical points of a polynomial

In this paper, we introduce a new type of companion matrices, namely, D-companion matrices. By using these D-companion matrices, we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial. In fact, this new approach will allow us to prove quite a number of new as well as known results on this topic. For example, we prove some results on the majorization of the critical points of a polynomial by its zeros. In particular, we give a different proof of a recent result of Gerhard Schmeisser on this topic. The same method allows us to prove a higher order Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma.