Real orthogonal polynomials in frequency analysis (original) (raw)

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.

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.

A new presentation of orthogonal polynomials with applications to their computation

Numerical Algorithms, 1991

In this paper a new presentation of orthogonal polynomials is given. It is based on the introduction of two auxiliary sequences of arbitrary monic polynomials and it leads to a very simple derivation of the usual determinantal formulae for orthogonal polynomials and of their recurrence relations either in the definite or in the indefinite case. New expressions for the coefficients of these recurrence relations are obtained and they are compared to the usual ones from the point of view of their numerical stability. The qd-algorithm is also recovered very easily.

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.

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.

New orthogonal polynomials for speech signal and image processing

IET Signal Processing, 2012

This study introduces a new set of orthogonal polynomials and moments and the set's application in signal and image processing. This polynomial is derived from two well-known orthogonal polynomials: the Tchebichef and Krawtchouk polynomials. This study attempts to present the following: (i) the mathematical and theoretical frameworks for the definition of this polynomial including the modelling of signals with the various analytical properties it contains, as well as, recurrence relations and transform equations that need to be addressed; and (ii) the results of empirical tests that compare the representational capabilities of this polynomial with those of the more traditional Tchebichef and Krawtchouk polynomials using speech and image signals from different databases. This study attempts to demonstrate that the proposed polynomials can be applied in the field of signal and image processing because of the promising properties of this polynomial especially in its localisation and energy compaction capabilities.

A few remarks on orthogonal polynomials

2014

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials { p_n} _n≥ 0 that are orthogonal with respect to this distribution, coefficients of expansion of x^n in the series of p_j, j≤ n, two sequences of coefficients of the 3-term recurrence of the family of { p_n} _n≥ 0, the so called "linearization coefficients" i.e. coefficients of expansion of p_j, j≤ m+n. Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials { p_n} _n≥ 0, we express with their help: coefficients of the power series expansion of p_n, coefficients of expansion of x^n in the series of p_j, j≤ n, moments of the distribution that makes polynomials { p_n} _n≥ 0 orthogonal. Further having two different families of orthogonal polynomials { p_n} _n≥ 0 and { q_n} _n≥ 0 and knowing for each of them sequences of the 3-term recurren...

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle

Acta Mathematica Hungarica - ACTA MATH HUNG, 2002

Given a probability measure μ on the unit circle T, we study para-orthogonal polynomials Bn(.,w) (with fixed w ∈ T) and their zeros which are known to lie on the unit circle. We focus on the properties of zeros akin to the well known properties of zeros of orthogonal polynomials on the real line, such as alternation, separation and asymptotic distribution. We also estimate the distance between the consecutive zeros and examine the property of the support of μ to attract zeros of para-orthogonal polynomials.