“Christoffel–Darboux formula”的意思、由来-开放百科全书

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Christoffel–Darboux formula

释义

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References

In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by {{harvs|txt|authorlink=Elwin Bruno Christoffel|first=Elwin Bruno|last= Christoffel|year=1858}} and {{harvs|txt|authorlink=Jean Gaston Darboux|first=Jean Gaston|last= Darboux|year=1878}}. It states that

where f_j(x) is the jth term of a set of orthogonal polynomials of squared norm h_j and leading coefficient k_j.

There is also a "confluent form" of this identity:

References

{{Citation | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=Cambridge University Press | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6 | mr=1688958 | year=1999 | volume=71}}
{{Citation | last1=Christoffel | first1=E. B. | title=Über die Gaußische Quadratur und eine Verallgemeinerung derselben. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002150239 | language=German | doi=10.1515/crll.1858.55.61 | year=1858 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | volume=55 | pages=61–82}}
{{Citation | last1=Darboux | first1=Gaston | title=Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série | language=French | jfm=10.0279.01 | year=1878 | journal=Journal de Mathématiques Pures et Appliquées | volume=4 | pages=5–56, 377–416}}
{{Citation | last1=Abramowitz | first1=Milton | last2=Stegun | first2=Irene A. | title=Handbook of Mathematical Functions | publisher=Dover Publications, Inc., New York | year=1972 | page=785, Eq. 22.12.1}}
{{Citation | last1=Olver | first1= Frank W. J. | last2=Lozier | first2=Daniel W. | last3=Boisvert | first3=Ronald F. | last4=Clark | first4=Charles W. | title=NIST Handbook of Mathematical Functions | year=2010 | isbn=978-0-521-19225-5 | publisher=Cambridge University Press | url=http://www.cambridge.org/9780521140638 | website=NIST Digital Library of Mathematical Functions | page=438, Eqs. 18.2.12 and 18.2.13}} (Hardback, {{ISBN|978-0-521-14063-8}} Paperback)

2 : Orthogonal polynomials|Functional analysis

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References