FW: Lebesgue functions of Legendre and Gegenbauer expansions

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

-----Original Message-----
From: opsftalk@nist.gov [mailto:opsftalk@nist.gov]On Behalf Of
kayumov@r66.ru
Sent: Monday, June 06, 2005 9:53 AM
Sender: <kayumov@r66.ru>
Subject: Lebesgue functions of Legendre and Gegenbauer expansions


Please forgive my disturbing you again with a question on orthogonal
polynomials.

In articles on Lebesgue constants for Legendre and Gegenbauer expansions
the following result is frequently cited or implied: that the Lebesgue
function attains its maximum value (which is thus the corresponding
Lebesgue constant) at the endpoints of the interval.

I am very much interested in understanding the idea behind the proof.
However, one is usually referred to the original article by Gronwall
(Uber die Laplacesche Reihe, Mathematische Annalen, 1913, 74) for
Legendre expansions and to articles by Kogbetliantz (in the 1910-1920s
in Comptes Rendus de l'Academie des Sciences Francaise) for Gegenbauer
expansions.

Neither one of these is available at my Institute here in Russia. I was
also unable to find a more recent exposition of the proof in any of the
standard references on orthogonal polynomials available to me (such as
Szego, Bateman and Erdelyi, Suetin, Nikiforov and Uvarov).

So does anyone know where I could possibly find a more recent discussion
of the result? Or perhaps someone would be so kind as to briefly state
the idea behind the proof?

Thank you in advance for any help. (My e-mail address is kayumov@r66.ru
or alexander_kayumov@yahoo.com.)

Very sincerely,
Alexander Kayumov,
Institute of Mathematics and Mechanics,
Ekaterinburg, Russia.