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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On an extremal problem


Author: Paul G. Nevai
Journal: Proc. Amer. Math. Soc. 74 (1979), 301-306
MSC: Primary 42C05; Secondary 41A05
MathSciNet review: 524305
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Abstract: Let $ X = ({x_1},{x_2}, \ldots ,{x_N}),f:{\mathbf{R}} \to {\mathbf{C}}$ and let $ {{\mathbf{P}}_n}$ be the class of polynomials of degree at most n. The generalized Christoffel function $ {\Lambda _n}$ corresponding to the measure $ d\alpha $ is defined by

$\displaystyle {\Lambda _n}(X;f,N,d\alpha ) = \mathop {\min }\limits_{\begin{arr... ... \\ \end{array} } \int_{ - \infty }^\infty {\vert\pi (t){\vert^2}d\alpha (t).} $

It is shown that if $ \alpha $ satisfies some rather weak conditions then $ {\lim _{n \to \infty }}n{\Lambda _n}(X;f,N,d\alpha )$ exists and the limit is also evaluated.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0524305-8
PII: S 0002-9939(1979)0524305-8
Keywords: Orthogonal polynomials, Christoffel functions
Article copyright: © Copyright 1979 American Mathematical Society