Publication: Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality
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2008-10-15
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Elsevier
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Abstract
We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same $n$th root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e<sup>−φ(x)</sup>, giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.
Description
9 pages, no figures.-- MSC2000 code: 33C45.
MR#: MR2431543 (2009g:41009)
Zbl#: Zbl 1155.33006
MR#: MR2431543 (2009g:41009)
Zbl#: Zbl 1155.33006
Keywords
Logarithmic potential theory, Orthogonal polynomials, Zero location, Asymptotic behavior
Bibliographic citation
Journal of Mathematical Analysis and Applications, 2008, vol. 346, n. 2, p. 480-488