On some classical problems concerning $L_{\infty}$-extremal polynomials with constraints

Details On some classical problems concerning $L_{\infty}$-extremal polynomials with constraints On some classical problems concerning $L_{\infty}$-extremal polynomials with constraints

2010-01-04

arxiv.org/abs/1001.0469v1

Mathematics

Classical Analysis and ODEs

The last modifications and corrections of this manuscript were done by the author in the two months preceding this passing awa

Scientific paper

First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 \pi].$ We show that the minimal polynomial is on $[0,2 \pi]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $\tau_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linear functionals on the space of trigonometric polynomials is determined. As a consequence a conjecture by Clenshaw from the sixties on the behavior of the ratio of the truncated Fourier series and the minimum deviation is proved.

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