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Topic: Orthogonal polynomials

	Orthogonal polynomials - Wikipedia, the free encyclopedia
		In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p
		The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Stieltjes.
		The simplest orthogonal polynomials are the Legendre polynomials, for which the interval of orthogonality is [−1, 1] and the weight function is simply 1:
	en.wikipedia.org /wiki/Orthogonal_polynomials (2218 words)

	Orthogonality - Wikipedia, the free encyclopedia
		In 4D the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.
		The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from −1 to 1.
		The Laguerre polynomials are orthogonal with respect to the exponential distribution.
	en.wikipedia.org /wiki/Orthogonal (1257 words)

	PlanetMath: orthogonal
		This is where the use of “orthogonal” in orthogonal lines, orthogonal circles, and other geometric terms come from.
		In the realm of linear algebra, two vectors are orthogonal when their dot product is zero, which gave rise a generalization of two vectors on some inner product space (not necessarily dot product) being orthogonal when their inner product is zero.
		This is version 9 of orthogonal, born on 2002-01-04, modified 2005-03-19.
	planetmath.org /encyclopedia/Orthogonal.html (217 words)

	PlanetMath: orthogonal polynomials
		Orthogonal polynomials are obtained in the following way: define the scalar product.
		Orthogonal polynomials of successive orders can be expressed by a recurrence relation
		This is version 6 of orthogonal polynomials, born on 2002-01-04, modified 2005-04-24.
	planetmath.org /encyclopedia/OrthogonalPolynomials.html (287 words)

	[No title]
		Although for the known to author systems of orthogonal polynomials these functions are themselves polynomials in $x$ and ($c(x,n)$) in $n$ (see [\ref{Geron_in_Szego},\ref{Bochner},\ref{Brenke}] for related questions), this fact is not important for what follows.
		Now fix the system of polynomials $\{y_n(x)\}_{n=0}^\infty$ satisfying equations (1) for $n=0,1,\dots$ We further assume that these polynomials are orthogonal with a continuous weight function on a set $x\in\cal A$.
		If $a(x)$ and $b(x)$ are polynomials of no greater than the second and first degree, respectively, and $c(n)$ does not depend on $x$, then (1) is an equation of the hypergeometric type.
	www.ma.utexas.edu /mp_arc/papers/97-367 (943 words)

	Laguerre Polynomials
		The Laguerre polynomials are orthogonal on the interval from 0 to ∞ with respect to the weight function w(x) = e
		Orthogonality of the generalized Laguerre polynomials is expressed by the integral
		The Laguerre polynomials are really rather straightforward examples of orthogonal polynomials, and most of their properties can be derived from Rodrigues' formula.
	www.du.edu /~jcalvert/math/laguerre.htm (972 words)

	Partial Differential Equations and Bivariate Orthogonal Polynomials -- from Mathematica Information Center
		One approach, due to Krall and Sheffer in 1967 and pursued by others is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue.
		In contrast, our approach is to seek pairs of linear differential operators which, have joint eigenfunctions that are comprises of family of bivariate orthogonal polynomials.
		This approach entails the addition of some "normalizing" or "regularity" conditions which allow determination of a unique family of orthogonal polynomials.
	library.wolfram.com /infocenter/Articles/1816 (193 words)

	opolyls5.html
		We have seen that the classical orthogonal polynomials can overcome the ill-conditioning in the least-squares problem, but an even better approach is to use discrete orthogonal polynomials [4] instead of using the standard monomial or classical orthogonal polynomial basis functions.
		One of the advantages of the discrete orthogonal polynomial approach is that we can find a lower degree least-squares polynomial, by dropping the higher degree terms, whereas the other methods using the standard monomial basis functions or the classical orthogonal polynomials require a recomputation.
		The discrete orthogonal polynomial method finds all the least-squares fiited polynomials up to degree 40 for the data given in Example 1, using double precision arithmetic and produces acceptable results.
	www.math.sfu.ca /~gfee/opolyls/opolyls51.html (1838 words)

	Legendre Polynomials (Site not responding. Last check: 2007-11-06)
		Legendre (biography) developed a set of orthogonal polynomials that can be used to approximate various functions.
		The error term is orthogonal to the first n+1 Legendre polynomials, hence it is orthogonal to all polynomials of degree n or less.
		Expand (h+e).(h+e), remembering that h is orthogonal to every polynomial of degree n or less, hence h is orthogonal to e.
	www.mathreference.com /la,legpoly.html (360 words)

	PREPRINTS
		On the asymptotic expansion of the entropy of Gegenbauer polynomials, J.
		Orthogonal polynomials on the unit circle: a matrix Riemann-Hilbert approach with some applications, talk at 7th Annual Workshop on Applications and Generalizations of Complex Analysis, Universidade de Aveiro, Portugal, 4-5 de junio de 2004.
		Orthogonal Polynomials on the Unit Circle: the Riemann-Hilbert Perspective, invited talk at Workshop 6, “Special functions and orthogonal polynomials”, of the V Conference “Foundations of Computational Mathematics”, Santander (Spain), June 30 – July 9, 2005.
	www.ual.es /GruposInv/Tapo/english/publicen.htm (3039 words)

	General Orthogonal Polynomials - Cambridge University Press
		In this treatise, the authors present the general theory of orthogonal polynomials on the complex plane and several of its applications.
		The assumptions on the measure of orthogonality are general, the only restriction is that it has compact support on the complex plane.
		Orthogonal polynomials with this behaviour correspond to classical orthogonal polynomials in the general case, and many extremal properties of measures in mathematical analysis and approximation theory with this type of regularity turn out to be equivalent.
	www.cambridge.org /catalogue/catalogue.asp?ISBN=0521415349 (321 words)

	Hermite Polynomials
		Of course, the corresponding polynomials will be very similar, and one could be used as well as the other, with appropriate changes of variable.
		To prove this, simply express the exponential times the Hermite polynomial of larger order as an nth derivative using the Rodrigues formula, then integrate by parts until the polynomial of smaller order is differentiated to zero.
		The orthogonality can be used to expand an arbitrary function f(x) in a series of Hermite polynomials, in exactly the same way that a Fourier series is formed.
	www.du.edu /~jcalvert/math/hermite.htm (1147 words)

	Citebase - Orthogonal Polynomials
		Orthogonal polynomials on the unit circle are not discussed.
		Orthogonal polynomials with periodic recurrence coefficients, J. Approx.
		Orthogonal Polynomials 123 [79] B. Simon, Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line, J. Approx.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0512424 (1646 words)

	Inzell Lectures
		The SIAM Activity Group on Orthogonal Polynomials and Special Functions organized a series of summer schools with the aim of opening the field to young researchers.
		Holger Dette (RU Bochum) told us about "Canonical Moments, Orthogonal Polynomials with Applications to Statistics," a subject with a rather new development as may be seen from the recent monograph by the speaker and W.J. Studden: "The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis," Wiley 1997.
		Xu stressed several times in his lectures that the theory of orthogonal polynomials in several variables is still in its very beginning.
	bigwww.epfl.ch /publications/zucastell0501.html (881 words)

	The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
		Roelof Koekoek and René F. Swarttouw: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue.
		This revised version includes a description of all families of hypergeometric orthogonal polynomials appearing in the Askey-scheme (named after Richard A. Askey) and in the q-analogue of this scheme.
		Classification of hypergeometric orthogonal polynomials Classification of hypergeometric orthogonal polynomials
	aw.twi.tudelft.nl /~koekoek/askey.html (510 words)

	Formal Orthogonal Polynomials (Site not responding. Last check: 2007-11-06)
		Formal Orthogonal Polynomials The connection between formal orthogonal polynomials (FOP), a generalization of the usual ones, and the Lanczos' method for solving systems of linear equations is known since Lanczos own papers of 1950 and 1952.
		They can also be found in * E.L. Stiefel, Kernel polynomials in linear algebra and their numerical applications, in "Further Contributions to the Solution of Simultaneous Linear Equations and the Determina- tion of Eigenvalues", NBS Appl.
		More recently, the algebra of linear functionals on the space of polynomials was formalized by Maroni in a series of papers * P. Maroni, Sur quelques espaces de distributions qui sont des formes lin\'eaires sur l'espace vectoriel des polyn\^omes, in "Polyn\^omes Orthogonaux et Applications", C. Brezinski et al.
	www.csc.fi /math_topics/Mail/NANET95/msg00144.html (516 words)

	Fast Decompositions by Gegenbauer Family Orthogonal Polynomials -- from Mathematica Information Center
		The efficient expansions by classical orthogonal polynomials is a well known and hard solving problem for applied mathematics.
		At first sight there is no problem as Mathematica has a package for the fast generation of all known classical orthogonal polynomials and has a powerful package for numerical integration.
		The main reason of this phenomenon is high oscillation of all known functional orthogonal systems if the corresponding index is rather big.
	library.wolfram.com /infocenter/MathSource/4748 (149 words)

	FOURIER SERIES IN ORTHOGONAL POLYNOMIALS
		This book presents a systematic course on general orthogonal polynomials and Fourier series in orthogonal polynomials.
		The main subject of the book is Fourier series in general orthogonal polynomials.
		for experts in orthogonal polynomials, the most useful part of the book is the "Notes", where the author gives several precise additional comments on the results presented, on their origin and on their generalizations...
	www.worldscibooks.com /mathematics/4039.html (314 words)

	Applied Orthogonal Polynomials (Site not responding. Last check: 2007-11-06)
		Currently, the main focus is on orthogonal polynomials of Sobolev type, i.e., on polynomials orthogonal with respect to an inner product involving derivatives.
		Modified moment algorithms known from the theory of ordinary orthogonal polynomials are being extended, tested, and analyzed and so are Stieltjes-type algorithms.
		Other current research involves Stieltjes polynomials and related quadrature formulae, the computation of Turán quadrature rules based on the theory of s-orthogonality, and quadrature convergence of extended Lagrange interpolation.
	www.cs.purdue.edu /AnnualReports/95/AR95Book-87.html (112 words)

	Some Discrete Multiple Orthogonal Polynomials - Arvesu, Coussement, Van Assche (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: In this paper we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures.
		First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn [4] [6] [7].
		These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order di#erence equation, and an explicit expression from which the coe#cients...
	citeseer.ist.psu.edu /468432.html (322 words)

	Orthogonal Polynomials; Interdisciplinary Aspects (Site not responding. Last check: 2007-11-06)
		There have been several regularly scheduled meetings in North America, Europe, and Asia in the subjects of orthogonal polynomials, special functions and their applications over the past twenty years.
		At the summer meeting of the CMS held in Saskatoon in 2001 one of the sessions was devoted to integrable systems, inverse scattering theory and emerging links with the theory of orthogonal polynomials.
		There was a general consensus among the participants that there is a great need for a continuing dialog between different groups of users of the theory of orthogonal polynomials and the orthogonal polynomials theorists.
	www.pims.math.ca /birs/workshops/2004/04w5530 (239 words)

	Orthogonal Polynomials (Site not responding. Last check: 2007-11-06)
		Introduction: Classical orthogonal polynomials are defined for the intervals [-1,1], [0,\infty] or [-\infty,\infty].
		When doing actual computations with orthogonal polynomials, several numbers are of importance, of which the leading coefficient and the norm are the most prominent.
		Other useful hints: Several respondents made me realize how badly I phrased the question by gently reminding me that one should evaluate orthogonal polynomials by the recurrence formula, and series involving these polynomials by Clenshaw's algorithm.
	www.csc.fi /math_topics/Mail/NANET94/msg00758.html (554 words)

	[No title]
		These polynomials which had been missed by others are now called sieved polynomials and they have played an important role in a few developments of the general theory of orthogonal polynomials.
		The polynomials of Rogers have a very interesting combinatorial structure associated with them, and extensions of this to the Al-Salam-Carlitz polynomials is currently being developed.
		The polynomials mentioned above are very appropriately named, and these and some other results found by him will keep his name in the minds of many people who never had the joy of knowing him.
	www.math.yorku.ca /Who/Faculty/Muldoon/siamopsf/personal/waleed.html (1155 words)

	Maxima Manual - Orthogonal Polynomials
		(The Jacobi polynomials are actually defined for all a and b ; however, the Jacobi polynomial weight (1-x)^a(1+x)^b isn't integrable for a <= -1 or b <= -1.
		Many functions in specfun are computed as a special case of the Jacobi polynomials; they also enjoy the speed boost from the modedeclared version of jacobi.
		The ultraspherical polynomials are also known as Gegenbauer polynomials.
	www.ma.utexas.edu /maxima/maxima_16.html (1215 words)

	Fischer: Orthogonal Polynomials (Site not responding. Last check: 2007-11-06)
		How to generate unknown orthogonal polynomials out of known orthogonal polynomials, with Gene Golub, in Journal of Comp.
		From orthogonal polynomials to iteration schemes for linear systems: CG and CR revisited, Wavelets and allied topics, Pawan K. Jain et al (eds.), Narosa publishing house,New Dehli, India,225-247, 2001.
		Orthogonal polynomial wavelets, with Woula Themistoclakis, Numerical Algorithms 30, 37-58, 2002.
	www.math.mu-luebeck.de /fischer/bernd_orthogonal.html (68 words)

	Volume 4, Number 2
		Readers who are interested in classical orthogonal polynomials should probably first look at [Chihara], [NU], [Rainville], [Szego], or [Tricomi]; readers looking for applications in mathematical physics might consider [NU].
		L.: "Polynomials Orthogonal on a Circle and Interval." International Series of Monographs on Pure and Applied Mathematics, Vol.
		The preprint archive "Orthogonal polynomials and related special functions" maintained by Hans Haubold has got a new WWW address: ftp://unvie6.un.or.at/siam/opsf_new/00index.html See Topic #13 for more information on this archive.
	cio.nist.gov /esd/emaildir/lists/opsfnet/msg00047.html (3845 words)

	[No title]
		The summer school was mainly focused on orthogonal polynomials and their various applications.
		After a general introduction to the theory of orthogonal polynomials of one variable, he focused on conditions leading to non-negative linearization coefficients (called property (P)).
		Xu gave his version of the three-term recurrence relation with applications to a Christoffel-Darboux formula and to common zeros of the orthonormal polynomials of total degree N. There is no agreement as to which systems of orthogonal polynomials of several variables should be called classical.
	math.nist.gov /opsf/reports/inzell01.html (1200 words)

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