Analytic Theory of Polynomials (original) (raw)

2. Review of orthogonal polynomials

De Gruyter eBooks, 2017

Review of orthogonal polynomials 2.1 Introduction Developments and interests in orthogonal polynomials have seen continuous and great progress since their appearance. Orthogonal polynomials are connected with many mathematical, physical, engineering, and computer sciences topics, such as trigonometry, hypergeometric series, special and elliptic functions, continued fractions, interpolation, quantum mechanics, partial differential equations. They are also be found in scattering theory, automatic control, signal analysis, potential theory, approximation theory, and numerical analysis. Orthogonal polynomials are special polynomials that are orthogonal with respect to some special weights allowing them to satisfy some properties that are not generally fulfilled with other polynomials or functions. Such properties have made them wellknown candidates to resolve enormous problems in physics, probability, statistics and other fields. Since their origin in the early 19th century, orthogonal polynomials have formed a somehow classical topic related to Legendre polynomials, Stieltjes' continued fractions, and the work of Gauss, Jacobi, and Christoffel, which has been generalized by Chebyshev, Heine, Szegö, Markov, and others. The most popular orthogonal polynomials are Jacobi, Laguerre, Hermite polynomials, and their special relatives, such as Gegenbauer, Chebyshev, and Legendre polynomials. An extending family has been developed from the work of Wilson, inducing a special set of orthogonal polynomials known by his name, which generalizes the Jacobi class. This new family has given rise to other previously unknown sets of orthogonal polynomials, including Meixner Pollaczek, Hahn, and Askey polynomials. Orthogonal polynomials may also be classified according to the measure applied to define the orthogonality. In this context, we cite the class of discrete orthogonal polynomials that form a special case based on some discrete measure. The most common are Racah polynomials, Hahn polynomials, and their dual class, which in turn include Meixner, Krawtchouk, and Charlier polynomials. Already with the classification of orthogonal polynomials, one can distinguish circular and generally spherical orthogonal polynomials, which consists of some special sets related to measures supported by the circle or the sphere. One well-known class is composed of Rogers-Szegö polynomials on the unit circle and Zernike polynomials, which are related to the unit disk. Orthogonal polynomials, and especially classical ones, can generally be introduced by three principal methods. A first method is based on the Rodrigues formula which consists of introducing orthogonal polynomials as outputs of a derivation.

On two models of orthogonal polynomials and their applications

This contribution deals with some models of orthogonal polynomials as well as their applications in several areas of mathematics. Some new trends in the theory of orthogonal polynomials are summarized. In particular, we emphasize on two kinds of orthogonality, i.e., the standard orthogonality in the unit circle and a non standard one, which is called multi-orthogonality. Both have attracted the interest of researchers during the past ten years.

Classical orthogonal polynomials: dependence of parameters

Journal of Computational and Applied Mathematics, 2000

Most of the classical orthogonal polynomials (continuous, discrete and their q-analogues) can be considered as functions of several parameters ci. A systematic study of the variation, inÿnitesimal and ÿnite, of these polynomials Pn(x; ci) with respect to the parameters ci is proposed. A method to get recurrence relations for connection coe cients linking (@ r =@c r i)Pn(x; ci) to Pn(x; ci) is given and, in some situations, explicit expressions are obtained. This allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function. A basic theorem on the zeros of (@=@ci)Pn(x; ci) is also proved.

On moments of classical orthogonal polynomials

Journal of Mathematical Analysis and Applications, 2015

Dr. Mama Foupouagnigni for the continuous support of my Ph.D study and research, for their patience, motivation, enthusiasm, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. I could not have imagined having better advisors and mentors for my Ph.D study. I am grateful to Prof. Dr. Mama Foupouagnigni for enlightening me the first glance of research. My sincere thanks also go to Prof. Dr. Wolfram Koepf for offering me the opportunity to visit the University of Kassel where part of this work has been written.

The Associated Classical Orthogonal Polynomials

The associated orthogonal polynomials {p n (x; c)} are defined by the 3-term recurrence relation with coefficients A n , B n , C n for {p n (x)} with c = 0, replaced by A n+c , B n+c and C n+c , c being the association parameter. Starting with examples where such polynomials occur in a natural way some of the well-known theories of how to determine their measures of orthogonality are discussed. The highest level of the family of classical orthogonal polynomials, namely, the associated Askey-Wilson polynomials which were studied at length by Ismail and Rahman in 1991 is reviewed with special reference to various connected results that exist in the literature.

Orthogonal polynomials on the real line

Walter Gautschi, Volume 2, 2013

In about two dozen papers, Walter Gautschi developed the so-called constructive theory of orthogonal polynomials on R, including effective algorithms for numerically generating orthogonal polynomials, a detailed stability analysis of such algorithms as well as several new applications of orthogonal polynomials. Furthermore, he provided software necessary for implementing these algorithms (see Section 23, Let P be the space of real polynomials and P n ⊂ P the space of polynomials of degree at most n. Suppose dµ(t) is a positive measure on R with finite or unbounded support, for which all moments µ k = R t k dµ(t) exist and are finite, and µ 0 > 0. Then the inner product (p, q) = R p(t)q(t)dµ(t) is well defined for any polynomials p, q ∈ P and gives rise to a unique system of monic orthogonal polynomials π k (•) = π k (• ; dµ); that is, π k (t) ≡ π k (t; dµ) = t k + terms of lower degree, k = 0, 1,. .. , and (π k , π n) = ||π n || 2 δ kn = 0, n ̸ = k, ||π n || 2 , n = k. 11.1. Three-term recurrence relation Because of the property (tp, q) = (p, tq), these polynomials satisfy a three-term recurrence relation π k+1 (t) = (t − α k)π k (t) − β k π k−1 (t), k = 0, 1, 2. .. , (11.1) Vol. 3) and applications.

A Unified Approach to Computing the Zeros of Classical Orthogonal Polynomials

2021

The authors present a unified method for calculating the zeros of the classical orthogonal polynomials based upon the electrostatic interpretation and its connection to the energy minimization problem. Examples are given with error estimates for three cases of the Jacobi polynomials, three cases of the Laguerre polynomials, and the Hermite polynomials. In the case of the Chebyshev polynomials, exact errors are given.

Real orthogonal polynomials in frequency analysis

Mathematics of Computation, 2004

We study the use of para-orthogonal polynomials in solving the frequency analysis problem. Through a transformation of Delsarte and Genin, we present an approach for the frequency analysis by using the zeros and Christoffel numbers of polynomials orthogonal on the real line. This leads to a simple and fast algorithm for the estimation of frequencies. We also provide a new method, faster than the Levinson algorithm, for the determination of the reflection coefficients of the corresponding real Szegő polynomials from the given moments.