Orthogonal polynomials

By Tom H. Koornwinder, Roderick S. C. Wong, R. Koekoek, Rene F. Swarttouw. NIST Handbook of Mathematical Functions, 2005

1,757 citations recorded by the source corpus.

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ee4204e5c0d58006735508e13762e7a6f3491138

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In Applied Mathematics

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Citations in the corpus

On the Clifford-Fourier transform (2010, methodology): “In the sequel we will often use the following well-known properties of Gegenbauer polynomials (see [25]):”
√n-consistent parameter estimation for systems of ordinary differential equations : bypassing numerical integration via smoothing (2010, neutral): “7 in Szegö (1975). Our fourth order kernel took the form”
A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations (2009, neutral): “SOME PROPERTIES OF JACOBI POLYNOMIALS The Jacobi polynomials associated with the real parameters (α > −1, β > −1) (see, [5,40]), are a sequence of polynomials P (α,β) n (x)(n = 0, 1, 2, .”
Entire extensions and exponential decay for semilinear elliptic equations (2010, background): “• When h is odd, for ρ = − 2( h + 1) N − h or else ρ = − 2( h + 1) N − h − 2 , N ∈ Z , the solution in L 2 ( R ) of (5.9) is of the form u ( x, ρ ) = exp[ − x h +1 / ( h + 1)] P ρ ( x ) (5.12) where P ρ ( x ) is a polynomial, cf. [26].”
Electrostatic models for zeros of polynomials: Old, new, and some open problems (2005, background): “There uniqueness of the global minimum is not obvious, but there is an elegant proof in [63], based on the inequality between the arithmetic and geometric means.”
MULTIPLICATIVE RENORMALIZATION AND GENERATING FUNCTIONS II (2003, background): “INTRODUCTION The theory of orthogonal polynomials has a long history (see for example [7] [8] [11] [13] and the references therein) with a wide range of applications.”
Analytic Combinatorics (2009, background): “(Historically, the theory of orthogonal polynomials evolved from the theory of continued f ractions, before living a life of its own; see [118, 343, 563] for its many facets.”
Basic Complex Analysis: A Comprehensive Course in Analysis, Part 2A (2015, methodology): “For the use of the Joukowski map in the theory of orthogonal polynomials, see [550, 525].”
Polynomial approximation on the sphere using scattered data (2008, neutral): “17) and [10] ∫ 1 −1 R ( q2 , q 2−1) n (x)R ( q2 , q 2−1) m (x)(1− x) q 2 (1 + x) q 2−1 dx = 2 [ Γ ( q 2 + 1 ) 2n + q ]2 n! (n + q − 1)! δn,m, we get the following orthogonality relation ∫ 1 −1 Kn(x)Km(x)(1− x) q 2 (1 + x) q 2−1 dx = 2−q+1 [ Γ ( q 2 )]2 (n + q − 1)! n! δn,m.”
Spectra of Graphs (2011, methodology): “Then the fj are orthogonal polynomials, and therefore they satisfy a 3-term recurrence relation (see Szegő [352], p.”
Quantum Mechanics for Mathematicians (2008, background): “11, which can be found in monograph [Sze75], is an example of semi-classical asymptotics, and Problem 3.”
Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type (2012, background): “Introduction to orthogonal polynomials on R The main concepts in the theory of orthogonal polynomials can be found in [16], [1], [5], [10].”