积分的正交多项式近似

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1、ElectronicTransactionsonNumericalAnalysis.ETNAVolume19,pp.58-83,2005.KentStateUniversityCopyrightÓ2005,KentStateUniversity.etna@mcs.kent.eduISSN1068-9613.ASYMPTOTICAPPROXIMATIONSOFINTEGRALS:ANINTRODUCTION,WITHRECENTDEVELOPMENTSANDAPPLICATIONSTOORTHOGONALPOLYNOMIALSCHELOFERREIRA
,JOSE´L.LO

2、PEZ´,ESMERALDAMAINAR,ANDNICOM.TEMMEAbstract.Intherstpartwediscusstheconceptofasymptoticexpansionanditsimportanceinapplications.Wefocusourattentiononspecialfunctionsdenedthroughintegralsandconsidertheirapproximationbymeansofasymptoticexpansions.Weexplainthegeneralideasofthetheoryofasym

3、ptoticexpansionsofintegralsanddescribetwoclassicalmethods(Watson'slemmaandthesaddlepointmethod)andmodernmethods(distributionalmethods).Inthesecondpartweapplytheseideastoapproximate(inanasymptoticsense)polynomialsoftheAskeytableintermsofsimplerpolynomialsoftheAskeytable.Weconsidertwodiffere

4、nttypesofasymptoticexpansionsthathavebeenrecentlydeveloped:i)someparameterofthepolynomialislargeorii)thedegree(andperhapsthevariabletoo)ofthepolynomialislarge.Foreachsituationweemployadifferentasymptoticmethod.Intherstcaseweusethetechniqueof“matchingofthegeneratingfunctionsattheorigin”.In

5、thesecondoneweemployamodiedversionofthesaddlepointmethodtogetherwiththetheoryoftwo-pointTaylorexpansions.Intherstcasetheasymptoticexpansionresultsinanitesumofpolynomials.Inthesecondonetheasymptoticexpansionisaconvergentinniteseriesofpolynomials.Weconcludethepaperwithinformationonotherr

6、ecentdevelopmentsintheresearchonasymptoticexpansionsofintegrals.Keywords.Asymptoticexpansionsofintegrals,asymptoticsoforthogonalpolynomials.AMSsubjectclassications.41A60,33C65.1.Introduction.Asymptoticanalysisisausefulmathematicaltoolwhichprovidesan-alyticalinsightandnumericalinformationa

7、boutthesolutionsofcomplicatedproblemsinappliedmathematics,engineering,physicsandmanyothersciences,whichrequireamathe-maticalframeworkfordescribingandmodelingscienticproblems.Someexamplesofthepowerofthistheoryarethefollowing:TheStirlingformulaforthefactorial: