共查询到20条相似文献,搜索用时 46 毫秒
1.
Wolfgang Erb 《Applied mathematics and computation》2011,217(9):4771-4780
We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball. 相似文献
2.
The theorems of Erd
s and Turán mentioned in the title are concerned with the distribution of zeros of a monic polynomial with known uniform norm along the unit interval or the unit disk. Recently, Blatt and Grothmann (Const. Approx.7(1991), 19–47), Grothmann (“Interpolation Points and Zeros of Polynomials in Approximation Theory,” Habilitationsschrift, Katholische Universität Eichstätt, 1992), and Andrievskii and Blatt (J. Approx. Theory88(1977), 109–134) established corresponding results for polynomials, considered on a system of sufficiently smooth Jordan curves and arcs or piecewise smooth curves and arcs. We extend some of these results to polynomials with known uniform norm along an arbitrary quasiconformal curve or arc. As applications, estimates for the distribution of the zeros of best uniform approximants, values of orthogonal polynomials, and zeros of Bieberbach polynomials and their derivatives are obtained. We also give a negative answer to one conjecture of Eiermann and Stahl (“Zeros of orthogonal polynomials on regularN-gons,” in Lecture Notes in Math.1574(1994), 187–189). 相似文献
3.
Herbert Stahl 《Constructive Approximation》2006,23(2):121-164
The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e}
polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$.
The polynomials are defined by the relation
$$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$
and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture
of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials
$p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a
precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials
$p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results
obtained in [17]. 相似文献
4.
Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros
except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives
is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations
of incomplete polynomials is also given. 相似文献
5.
George Csordas Marios Charalambides Fabian Waleffe 《Proceedings of the American Mathematical Society》2005,133(12):3551-3560
Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polynomials. Stability-preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials.
6.
Fuad Kittaneh 《Proceedings of the American Mathematical Society》2007,135(3):659-664
We apply some eigenvalue inequalities to the real parts of the Frobenius companion matrices of monic polynomials to establish new bounds and a majorization for the real parts of the zeros of these polynomials.
7.
In this paper, we develop methods for establishing improved bounds on the moduli of the zeros of complex and real polynomials. Specific (lacunary) as well as arbitrary polynomials are considered. The methods are applied to specific polynomials by way of example. Finally, we evaluate the quality of some bounds numerically. 相似文献
8.
9.
In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.Research of Juan José Moreno Balcázar was partially supported by Ministerio de Educación y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM 229 and FQM 481). 相似文献
10.
Predrag Rajković Franz Hinterleitner Sladjana Marinković 《Mathematical Methods in the Applied Sciences》2016,39(9):2358-2367
We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second‐order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite‐type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues‐type formula and a four‐term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
11.
Á. Pintér 《Journal of Mathematical Analysis and Applications》2002,270(1):303-305
It is given an upper bound for the number of simple and distinct zeros of the polynomial f+g, where f and g are relatively prime polynomials with complex coefficients. 相似文献
12.
Zélia da Rocha 《Journal of Difference Equations and Applications》2019,25(1):97-118
Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation. 相似文献
13.
We consider a modification of the gamma distribution by adding a discrete measure Support in the point x = 0. We study some properties of the polynomials orthogonal with respect to such measures [1]. In particular, we deduce the second order differential to'1ttatiolt and the three term recurrence relation which such polynomials satisfy as well as, for large n. the behaviour of their zeros. 相似文献
14.
The aim of this paper is to investigate some general properties of common zeros of orthogonal polynomials in two variables for any given region D⊂R2 from a view point of invariant factor. An important result is shown that if X0 is a common zero of all the orthogonal polynomials of degree k then the intersection of any line passing through X0 and D is not empty. This result can be used to settle the problem of location of common zeros of orthogonal polynomials in two variables. The main result of the paper can be considered as an extension of the univariate case. 相似文献
15.
We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros. 相似文献
16.
17.
André Draux 《Numerical Algorithms》2000,24(1-2):31-58
Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially
in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal
lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant
in Markov-Bernstein inequalities in L
2 for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of
the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new
sufficient condition is given in order that a polynomial has some complex zeros.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
18.
Martin Hanke 《Numerical Algorithms》1996,11(1):203-213
The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions. 相似文献
19.
We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous oneperturbed by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied. 相似文献
20.
Hong-Yong Wang 《Journal of Mathematical Analysis and Applications》2010,369(2):453-472
We study a class of sieved Pollaczek polynomials defined by a second-order difference equation (three-term recurrence relation). The measure of orthogonality is determined by using the Markov theorem and the Perron-Stieltjes inversion formula, and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points. Uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point αn and a critical point βn, involving respectively the Airy function Ai, and . Darboux's method, the method of steepest descents, and various uniform asymptotic techniques such as cubic transformations are used to derive the results. Asymptotic formulas for the least zeros, the largest zeros, and the zeros on both sides of βn are also obtained. Several numerical examples are provided to compare the approximate zeros with the true values. 相似文献