Orthogonal polynomials

Previous article Next article On Asymptotic Average Properties of Zeros of Orthogonal PolynomialsPaul G. Nevai and Jesus S. DehesaPaul G. Nevai and Jesus S. Dehesahttps://doi.org/10.1137/0510107PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractDistribution of zeros of orthogonal polynomials is investigated by means of the coefficients of the three-term recurrence relation which generates the orthogonal polynomials.[1A] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xxvi+302, Bateman Manuscript Project MR0058756 0051.30303 Google Scholar[1B] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xvii+396, Bateman Manuscript Project MR0058756 0052.29502 Google Scholar[2] G. Freud, Orthogonal Polynomials, Pergamon Press, New York, 1971 Google Scholar[3] Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc., 18 (1979), v+185 MR519926 0638.42023 ISIGoogle Scholar[4] David Dickinson, On certain polynomials associated with orthogonal polynomials, Boll. Un. Mat. Ital. (3), 13 (1958), 116–124 MR0104000 0100.06503 Google Scholar[5] J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzengenden Function, J. London Math. Soc., 9 (1934), 6–13 0008.16205 CrossrefGoogle Scholar[6] L. Carlitz, Some orthogonal polynomials related to elliptic functions, Duke Math. J., 27 (1960), 443–459 10.1215/S0012-7094-60-02742-3 MR0123030 0096.26903 CrossrefISIGoogle Scholar[7] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, 1978xii+249 MR0481884 0389.33008 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Complete solutions of inverse quantum orthogonal equivalence classesExamples and Counterexamples, Vol. 1 | 1 Nov 2021 Cross Ref Relative asymptotics for general orthogonal polynomialsMichigan Mathematical Journal, Vol. 66, No. 1 | 1 Mar 2017 Cross Ref The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relationsSbornik: Mathematics, Vol. 205, No. 12 | 16 February 2015 Cross Ref Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional ApproximationAkil Narayan and John D. 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Y. Goh and Jet WimpSIAM Journal on Mathematical Analysis, Vol. 25, No. 2 | 1 August 2006AbstractPDF (674 KB)Zeros of Orthogonal and Biorthogonal Polynomials: Some Old, Some NewNonlinear Numerical Methods and Rational Approximation II | 1 Jan 1994 Cross Ref An update on orthogonal polynomials and weighted approximation on the real lineActa Applicandae Mathematicae, Vol. 33, No. 2-3 | 1 Dec 1993 Cross Ref On a new characterization of the classical orthogonal polynomialsJournal of Approximation Theory, Vol. 71, No. 1 | 1 Oct 1992 Cross Ref WKB (Liouville-Green) analysis of second order difference equations and applicationsJournal of Approximation Theory, Vol. 69, No. 3 | 1 Jun 1992 Cross Ref N TH ROOT ASYMPTOTICS FOR EXTREMAL ERRORS ASSOCIATED WITH SLOWLY DECREASING WEIGHTSQuaestiones Mathematicae, Vol. 14, No. 3 | 1 Jul 1991 Cross Ref The one-quarter class of orthogonal polynomialsRocky Mountain Journal of Mathematics, Vol. 21, No. 1 | 1 Mar 1991 Cross Ref Asymptotic Behaviour for Wall Polynomials and the Addition Formula for Little q-Legendre PolynomialsWalter Van Assche and Tom H. KoornwinderSIAM Journal on Mathematical Analysis, Vol. 22, No. 1 | 1 August 2006AbstractPDF (792 KB)Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficientsJournal of Approximation Theory, Vol. 62, No. 1 | 1 Jul 1990 Cross Ref Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weightsConstructive Approximation, Vol. 5, No. 1 | 1 Dec 1989 Cross Ref Uniform and mean approximation by certain weighted polynomials, with applicationsConstructive Approximation, Vol. 4, No. 1 | 1 Dec 1988 Cross Ref On the asymptotic distribution of eigenvalues of banded matricesConstructive Approximation, Vol. 4, No. 1 | 1 Dec 1988 Cross Ref Asymptotic properties of orthogonal polynomials from their recurrence formula, IIJournal of Approximation Theory, Vol. 52, No. 3 | 1 Mar 1988 Cross Ref Rational approximations, orthogonal polynomials and equilibrium distributionsOrthogonal Polynomials and their Applications | 28 September 2006 Cross Ref Eigenvalues of Toeplitz matrices associated with orthogonal polynomialsJournal of Approximation Theory, Vol. 51, No. 4 | 1 Dec 1987 Cross Ref The ratio of q-like orthogonal polynomialsJournal of Mathematical Analysis and Applications, Vol. 128, No. 2 | 1 Dec 1987 Cross Ref A survey of general orthogonal polynomials for weights on finite and infinite intervalsActa Applicandae Mathematicae, Vol. 10, No. 3 | 1 Nov 1987 Cross Ref On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomialsJournal of Approximation Theory, Vol. 50, No. 3 | 1 Jul 1987 Cross Ref Quantum systems with a common density of levels. 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A case studyJournal of Approximation Theory, Vol. 48, No. 1 | 1 Sep 1986 Cross Ref Asymptotics for the Greatest Zeros of Orthogonal PolynomialsAttila MáTé, Paul Nevai, and Vilmos TotikSIAM Journal on Mathematical Analysis, Vol. 17, No. 3 | 1 August 2006AbstractPDF (547 KB)On Freud's equations for exponential weightsJournal of Approximation Theory, Vol. 46, No. 1 | 1 Jan 1986 Cross Ref Quantum systems with a common density of levelsPhysics Letters A, Vol. 113, No. 9 | 1 Jan 1986 Cross Ref Asymptotic properties of orthogonal polynomials from their recurrence formula, IJournal of Approximation Theory, Vol. 44, No. 3 | 1 Jul 1985 Cross Ref On the polynomial solutions of ordinary differential equations of the fourth orderJournal of Mathematical Physics, Vol. 26, No. 7 | 1 Jul 1985 Cross Ref Quantum systems with uniform- and regular-level-energy behaviorsPhysical Review A, Vol. 32, No. 1 | 1 July 1985 Cross Ref A proof of Freud's conjecture about the orthogonal polynomials related to |x|ρexp(−x2m), for integer m.Polynômes Orthogonaux et Applications | 16 September 2006 Cross Ref Two of My Favorite Ways of Obtaining Asymptotics for Orthogonal PolynomialsAnniversary Volume on Approximation Theory and Functional Analysis | 1 Jan 1984 Cross Ref On the largest zeroes of orthogonal polynomials for certain weightsMathematics of Computation, Vol. 41, No. 163 | 1 January 1983 Cross Ref Lanczos method of tridiagonalization, Jacobi matrices and physicsJournal of Computational and Applied Mathematics, Vol. 7, No. 4 | 1 Dec 1981 Cross Ref Orthogonal polynomials in transport theoriesJournal of Physics A: Mathematical and General, Vol. 14, No. 2 | 1 January 1999 Cross Ref The eigenvalue density of rational Jacobi matrices. IILinear Algebra and its Applications, Vol. 33 | 1 Oct 1980 Cross Ref Volume 10, Issue 6| 1979SIAM Journal on Mathematical Analysis1095-1326 History Submitted:09 May 1978Published online:17 February 2012 InformationCopyright © 1979 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0510107Article page range:pp. 1184-1192ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg–Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram–Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.

Orthogonal matrix polynomials and applications

Given a controllable system described by ordinary or partial differential equations and an initial state, the problem of finding a set of bounded positional controls that transfer the initial state to another state, not necessarily an equilibrium point, in finite time is called the synthesis problem. In the present work, we consider a family of Brunovsky systems of dimension n. A family of bounded positional controls un(x) is developed to stabilize a given Brunovsky system in finite time. We employ orthogonal polynomials associated with a function distribution σ(τ, θ) defined for τ ∈ [0, +∞) and parameter θ > 0. The parameter θ is interpreted as a Korobov’s controllability function, θ = θ(x), which serves as a Lyapunovtype function. Utilizing θ(x), we construct the positional control un(x) = un(x, θ(x)). Our analysis is based on the foundational work “A general approach to the solution of the bounded control synthesis problem in a controllability problem”. Matematicheskii Sbornik, 151(4), 582–606 (1979) by Korobov, V. I, in which the controllability function method was proposed. This method has been applied to solve bounded finite-time stabilization problems in various control scenarios, such as the control of the wave equation, optimal control with mixed cost functions, and other applications. For the construction of the mentioned positional controls, we employ a member of a family of orthogonal polynomials on [0,∞). For orthogonal polynomials, we refer to “Orthogonal Polynomials”. American Mathematical Society, Providence, (1975) by G. Szego. We also rely on the work “On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials”. Linear Algebra and its Applications, 476, 56–84 (2015) by Choque Rivero, A. E. The results in the present work extend and develop the findings presented in the conference paper “Bounded finite-time stabilizing controls via orthogonal polynomials”. 2018 IEEE International Autumn Meeting on Power, Electronics and Computing (ROPEC), Ixtapa, Mexico. –2018 by Choque-Rivero A. E., Orozco B. d. J. G.

A very general set of orthogonal polynomials with five free parameters is given explicitly, the orthogonality relation is proved and the three term recurrence relation is found.

A method is presented to compute the roots of complex orthogonal and kernel polynomials. An important application of complex kernel polynomials is the acceleration of iterative methods for the solution of nonsymmetric linear equations. In the real case, the roots of orthogonal polynomials coincide with the eigenvalues of the Jacobi matrix, a symmetric tridiagonal matrix obtained from the defining three-term recurrence relationship for the orthogonal polynomials. In the real case kernel polynomials are orthogonal. The Stieltjes procedure is an algorithm to compute the roots of orthogonal and kernel polynomials based on these facts. In the complex case, the Jacobi matrix generalizes to a Hessenberg matrix, the eigenvalues of which are roots of either orthogonal or kernel polynomials. The resulting algorithm generalizes the Stieltjes procedure. It may not be defined in the case of kernel polynomials, a consequence of the fact that they are orthogonal with respect to a nonpositive bilinear form. (Another consequence is that kernel polynomials need not be of exact degree.) A second algorithm that is always defined is presented for kernel polynomials. Numerical examples are described.

A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line

Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

Orthogonal polynomials, unless they are classical, require special techniques for their computation. One of the central problems is to generate the coefficients in the basic three-term recurrence relation they are known to satisfy. There are two general approaches for doing this: methods based on moment information, and discretization methods. In the former, one develops algorithms that take as input given moments, or modified moments, of the underlying measure and produce as output the desired recurrence coefficients. In theory, these algorithms yield exact answers. In practice, owing to rounding errors, the results are potentially inaccurate depending on the numerical condition of the mapping from the given moments (or modified moments) to the recurrence coefficients. A study of related condition numbers is therefore of practical interest. In contrast to moment-based algorithms, discretization methods are basically approximate methods: one approximates the underlying inner product by a discrete inner product and takes the recurrence coefficients of the corresponding discrete orthogonal polynomials to approximate those of the desired orthogonal polynomials. Finding discretizations that yield satisfactory rates of convergence requires a certain amount of skill and creativity on the part of the user, although general-purpose discretizations are available if all else fails. Other interesting problems have as objective the computation of new orthogonal polynomials out of old ones. If the measure of the new 1 orthogonal polynomials is the measure of the old ones multiplied by a rational function, one talks about modification of orthogonal polynomials and modification algorithms that carry out the transition from the old to the new orthogonal polynomials. This enters into a circle of ideas already investigated by Christoffel in the 1850s, but effective algorithms have been obtained only very recently. They require the computation of Cauchy integrals of orthogonal polynomials — another interesting computational problem. In the 1960s, a new type of orthogonal polynomials emerged — the so-called Sobolev orthogonal polynomials — which are based on inner products involving derivatives. Although they present their own computational challenges, moment-based algorithms and discretization methods are still two of the main stocks of the trade. The computation of zeros of Sobolev orthogonal polynomials is of particular interest in practice. An important application of orthogonal polynomials is to quadrature, specifically quadrature rules of the highest algebraic degree of exactness. Foremost among them is the Gaussian quadrature rule and its close relatives, the Gauss–Radau and Gauss–Lobatto rules. More recent extensions are due to Kronrod, who inserts n+1 new nodes into a given n-point Gauss formula, again optimally with respect to degree of exactness, and to Turan, who allows derivative terms to appear in the quadrature sum. When integrating functions having poles outside the interval of integration, quadrature rules of polynomial/rational degree of exactness are of interest. Poles inside the interval of integration give rise to Cauchy principal value integrals, which pose computational problems of their own. Interpreting Gaussian quadrature sums in terms of matrices allows interesting applications to the computation of matrix functionals. In the realm of approximation, orthogonal polynomials, especially discrete ones, find use in curve fitting, e.g. in the least squares approximation of discrete data. This indeed is the problem in which orthogonal polynomials (in substance if not in name) first appeared in the 1850s in work of Chebyshev. Sobolev orthogonal polynomials also had their origin in least squares approximation, when one tries to fit simultaneously functions together with some of their derivatives. Physically motivated are approximations by spline functions that preserve as many moments as possible. Interestingly, these also are related to orthogonal polynomials via Gauss and generalized Gauss-type quadrature formulae. Slowly convergent series whose sum can be expressed

It is well known that classes of polynomials in one variable defined by various extremality conditions play an extremely important role in complex analysis. Among these classes we find orthogonal polynomials (especially classical orthogonal polynomials expressed as hypergeometric polynomials) and polynomials least deviating from zero on a given continuum (Chebicheff polynomials). Orthogonal polynomials of the first and second kind appear as denominators and numerators of the Pade approximations to functions of classical analysis and satisfy familiar three-term linear recurrences. These polynomials were used repeatedly to study diophantine approximations of values of functions of classical analysis, especially exponential and logarithmic functions at rational points x = p/q [1], [2], [3], [4], [5]. The methods of Pade approximation in diophantine approximations are quite powerful and convenient to use, since they replace the problem of rational approximations to numbers with the approximations of functions. There are, however, arithmetic restrictions on rational approximations to functions if they are to be used for diophantine approximations. The main restriction on polynomials here is to have rational integer coefficients or rational coefficients with a controllable denominator. Such arithmetic restrictions transform a typical problem of classical analysis into an unusual mixture of arithmetic and analytic difficulties. For example, recurrences defining orthogonal polynomials must be of a special type to guarantee that their solutions will have hounded denominators. In this paper we consider various classes of polynomials generated by imposing arithmetic restrictions on classical approximation theory problems (orthogonal or Chebicheff polynomials).

A survey on orthogonal matrix polynomials satisfying second order differential equations

This paper gives some previously obtain results and contributions achieved in last fifteenth years on the topic of theory of orthogonal polynomials, i.e., orthogonal filters. This theory is based on new definitions and specific generalizations of orthogonal functions and polynomials, derived directly in complex domain. The main subject of this paper will be the possibility of some new applications of orthogonal polynomials in identification, modelling, signal processing and control of dynamical systems. Accordingly, the paper is divided into six sections. All chapters begin with a short mathematical background. In this paper we give some main results for classical, almost, improved almost, quasi-, generalized, and digital orthogonal polynomials.

The solution of several instances of the Schrodinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585–600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Grunbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461–84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.

In this paper, we study the theory of orthogonal trigonometric polynomials (OTPs). We obtain asymptotics of OTPs with positive and analytic weight functions by Riemann–Hilbert approach and find that they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, Christoffel–Darboux formula and some algebraic and asymptotic properties of zeros for orthogonal trigonometric polynomials.

Orthogonal polynomials over the interior of a unit circle are widely used in aberration theory and in describing ocular wavefront in ophthalmic applications. In optics, Zernike polynomials (ZPs) are commonly applied for the same purpose, and scaling their expansion coefficients to arbitrary aperture sizes is a useful technique to analyze systems with different pupil sizes. By employing the orthogonal Fourier-Mellin polynomials and their properties, a new formula is established based on the same techniques used to develop the scaled pupil sizes. The description by the orthogonal Fourier-Mellin polynomials for the aberration functions is better than that by the ZPs in terms of the wavefront reconstruction errors.