Re: FW: question on an integral involving a Jacobi polynomial

["Dan Lozier" <lozier@nist.gov>]

Sender: Martin Muldoon <muldoon@mathstat.yorku.ca>
Subject: Re: FW: question on an integral involving a Jacobi polynomial


You should look at the article by L. Lorch, The Lebesgue constants for
Jacobi series, I, Proc. Amer. Math. Soc. 10 (1959), 756-761, especially
the last part where your integral is given by the notation 4L(1,1,3,1)
and
is shown to be asymptotic to a multiple of n^(3/2) as n -> infty.
There are many more recent and precise results on related integrals but
I
don't find one that covers the parameters in your case.
Martin

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>
> -----Original Message-----
> From: opsftalk@nist.gov [mailto:opsftalk@nist.gov] On Behalf Of
> kayumov@r66.ru
> Sent: Saturday, May 21, 2005 9:54 AM
>
> Sender: <kayumov@r66.ru>
> Subject: question on an integral involving a Jacobi polynomial
>
>
> My name is Alexander Kayumov, I am a graduate student at the Institue
of
> Mathematics and Mechanics of the Russian Academy of Sciences
> (Ekaterinburg, Russia). In the course of my research on least-squares
> approximation by splines, I came upon the following problem.
>
> Consider the integral:
>
> \int_{-1}^1 |(1+t)P_n^{(1,2)}(t)| dt,
> where P^{(1,2)}_n(t) is a Jacobi polynomial with indices (1,2) and
> standard normalization.
>
> I need to find upper and lower estimates for this integral for all
n>=0
> (I presume it cannot be evaluated in terms of an explicit formula), or
> at least its asymptotic behaviour as n tends to infinity.
>
> I would extremely appreciate
> - any pointers to existing results, in case someone has already
tackled
> this problem, a similar or a more general one,
> - any hints or suggestions on how one might go about studying this
> integral, in case no one has done anything similar yet.
>
> Please forgive me for soliciting your help on this minor question and
> thank you in advance for any suggestions. (My e-mail address is
> kayumov@r66.ru or alexander_kayumov@yahoo.com.)
>
> Very sincerely,
> Alexander Kayumov.
>