On an extremal problem

Nevai, Paul G.

doi:10.1090/S0002-9939-1979-0524305-8

On an extremal problem
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by Paul G. Nevai PDF

Proc. Amer. Math. Soc. 74 (1979), 301-306 Request permission

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 \[ {\Lambda _n}(X;f,N,d\alpha ) = \min \limits _{\begin {array}{*{20}{c}} {\pi \in {{\mathbf {P}}_{n - 1}}} \\ {\pi ({x_i}) = f({x_i})} \\ {i = 1,2, \ldots ,N} \\ \end {array} } \int _{ - \infty }^\infty {|\pi (t){|^2}d\alpha (t).} \] 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

Orthogonal polynomials

Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR 0094840
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Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213

Orthogonal polynomials