共查询到20条相似文献,搜索用时 15 毫秒
1.
在单位圆周上考虑带特定权函数的正交多项式,利用Deift P.和Zhou X.所引进的关于振荡型Riemann-Hilbert问题的最速下降法,建立了这类正交多项式在整个复平面上的强渐近公式,发展和改进了一些经典结果. 相似文献
2.
In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993). 相似文献
3.
Antonio J. Duran 《Journal of Approximation Theory》1999,100(2):2239
Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind. 相似文献
4.
5.
This paper considers systems of Laguerre-type orthogonal polynomials for which the corresponding Jacobi matrices represent
unbounded self-adjoint operators which are bounded above or below. Under appropriate assumptions on the coefficient sequences
in the recursion formula, results are obtained on the uniform boundedness of the polynomials on bounded intervals, the absence
of eigenvalues for the corresponding operator, and the absolute continuity of the measure of orthogonality.
Date received: September 7, 1995. Date revised: April 17, 1996. 相似文献
6.
We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously
based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term
of order N
−2γ
where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the
pertinent literature.
The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the
location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity
at the rate N
−γ
, whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging
band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.
相似文献
7.
In the present paper we prove Szegő's asymptotic theorem for the orthogonal polynomials with respect to a Sobolev inner product
of the following type:
with μ
i
, i=0,···,p-1, finite positive Borel measures on [0,2π] and μ
p
a measure in the Szegő class. 相似文献
8.
A. Martinez-Finkelshtein K.T.-R. McLaughlin E.B. Saff 《Constructive Approximation》2006,24(3):319-363
We provide a representation in terms of certain canonical functions
for a sequence of polynomials orthogonal with respect to a weight
that is strictly positive and analytic on the unit circle. These
formulas yield a complete asymptotic expansion for these
polynomials, valid uniformly in the whole complex plane. As a
consequence, we obtain some results about the distribution of zeros
of these polynomials. The main technique is the steepest descent
analysis of Deift and Zhou, based on the matrix Riemann-Hilbert
characterization proposed by Fokas, Its, and Kitaev. 相似文献
9.
Wavelets Based on Orthogonal Polynomials 总被引:2,自引:0,他引:2
We present a unified approach for the construction of polynomial wavelets. Our main tool is orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.
10.
In this paper, we study the asymptotics of the Krawtchouk polynomials KnN(z;p,q) as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c ∈ (0, 1) as n →∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p;in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou. 相似文献
11.
Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit 总被引:1,自引:0,他引:1
We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples
by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis
Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction
via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical
experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery
rates are very similar.
相似文献
12.
P. López-Rodríguez 《Constructive Approximation》1999,15(1):135-151
We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of
V for which the matrix polynomials are dense in the corresponding
2
space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set.
May 20, 1997. Date revised: January 8, 1998. 相似文献
13.
Strong (or Szeg
-type) asymptotics for orthogonal polynomials with respect to a Sobolev inner product with general measures (the first measure is arbitrary and the second one is absolutely continuous and satisfying a smoothness condition) is obtained. Examples, illustrating the theorems proved, are presented. 相似文献
14.
Jacek Gilewicz Elie Leopold Andreas Ruffing Galliano Valent 《Constructive Approximation》2006,24(1):71-89
The orthogonal polynomials with recurrence relation
(λ,n +μn-z)Fn(z) = μn+1Fn+1(z)+λn-1Fn-1(z) with two kinds of cubic transition rates λn and μn, corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes
of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices
in the complex plane. 相似文献
15.
Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of lim n |Φ n (0)| and {lim n [ Φ n+1 (0)/ Φ n (0)] , where denotes the sequence of reflection coefficients. The limit periodic case, that is, when these limits exist for n = j mod k , j = 1, . . ., k , is also considered. December 27, 1996. Date revised: October 14, 1997. Date accepted: December 22, 1997. 相似文献
16.
E. Bourreau 《Acta Appl Math》2000,61(1-3):53-64
In the scalar case, computation of recurrence coefficients of polynomials orthogonal with respect to a nonnegative measure is done via the modified Chebyshev algorithm. Using the concept of matrix biorthogonality, we extend this algorithm to the vector case. 相似文献
17.
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
18.
The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order
differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix
W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of
some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical
functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer
parameters to obtain an immediate extension beyond these cases. 相似文献
19.
A. I. Aptekarev J. S. Dehesa A. Martínez-Finkelshtein R. Yáñez 《Constructive Approximation》2009,30(1):93-119
Given a nontrivial Borel measure on ℝ, let p
n
be the corresponding orthonormal polynomial of degree n whose zeros are λ
j
(n), j=1,…,n. Then for each j=1,…,n,
with
defines a discrete probability distribution. The Shannon entropy of the sequence {p
n
} is consequently defined as
In the case of Chebyshev polynomials of the first and second kinds, an explicit and closed formula for
is obtained, revealing interesting connections with number theory. In addition, several results of numerical computations
exemplifying the behavior of
for other families are presented.
相似文献
20.
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