Extremizers for the Rogosinski-Szeg\"o estimate of the second coefficient in nonnegative sine polynomials

TL;DR

This paper identifies extremizers for the Rogosinski-Szeg"o estimate of the second coefficient in nonnegative sine polynomials, providing explicit extremal polynomials and proving their uniqueness using advanced algebraic and matrix techniques.

Contribution

It constructs explicit extremizers for the Rogosinski-Szeg"o bounds and proves their uniqueness within the class of nonnegative sine polynomials.

Findings

Explicit extremizers attain the bounds for the second coefficient.

Uniqueness of extremizers is established.

Uses advanced algebraic and matrix methods for proof.

Abstract

For the class of sine polynomials $b_1\sin t+b_2\sin2t+...+b_N\sin Nt,\; (b_N\not= 0),$ which are nonnegative on $(0,\pi)$, W. Rogosinski and G. Szeg\"o derived, among other things, exact bounds for $|b_2|$ via the Luk\'acs presentation of nonnegative algebraic polynomials and a variational type argument for exact bounds, but they did not find the extremizers. Within this algebraic framework, we construct explicit polynomials which attain these bounds and prove their uniqueness. The proof uses the Fej\'er -Riesz representation of nonnegative trigonometric polynomials, a 7-band Toeplitz matrix of arbitrary finite dimension, and Chebyshev polynomials of the second kind and their derivatives.