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

TL;DR

This paper investigates classical extremal problems for $L_{ abla}$-norm constrained polynomials, revealing asymptotic behaviors, connections to Blaschke products, and proving a longstanding conjecture on Fourier series ratios.

Contribution

It establishes the asymptotic form of extremal trigonometric polynomials with given coefficients and proves Clenshaw's conjecture on Fourier series behavior.

Findings

Minimal polynomials asymptotically resemble a Blaschke product times a constant.

The constant is the largest singular value of a related Hankel matrix.

Clenshaw's conjecture on Fourier series ratio is proved.

Abstract

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.