Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] G. Freud, Orthogonal polynomials, Pergamon, New York, 1971.
  • [2] U. Grenander and G. Szegő, Toeplitz forms and their applications, Univ. of California Press, Berkeley, 1958. MR 0094840 (20:1349)
  • [3] U. Grenander and M. Rosenblatt, An extension of a theorem of G. Szegő and its application to the study of stochastic processes, Trans. Amer. Math. Soc. 76 (1954), 112-126. MR 0058902 (15:448e)
  • [4] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. (to appear). MR 519926 (80k:42025)
  • [5] G. Szegő, Orthogonal polynomials, Amer. Math. Soc.. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1967; 4th ed. 1975.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42C05, 41A05

Retrieve articles in all journals with MSC: 42C05, 41A05

Additional Information

Keywords: Orthogonal polynomials, Christoffel functions
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society