Extremizers for the Rogosinski-Szegö estimate of the second coefficient in nonnegative sine polynomials

TL;DR

This paper resolves the Rogosinski–Szegö problem for the second coefficient of typically real polynomials by constructing explicit, unique extremizers. The authors convert the analytic constraint Im$\{P(e^{it})\}\ge 0$ into a nonnegative cosine-polynomial problem via the Fejér–Riesz representation, then optimize two quadratic forms using a 7-band Toeplitz matrix pencil, yielding a clean eigenvalue characterization. The odd and even cases are treated separately, with closed-form extremizers expressed as sums of two rational functions and accompanied by compact representations of the associated nonnegative kernels; the bounds are $|a_2|\le 2\cos\frac{2\pi}{N+3}$ for odd $N$ and $|a_2|\le 2\cos\theta$ (where $\theta$ solves a root equation) for even $N$. The results not only recover the classical bounds but also provide explicit extremizers and their uniqueness, enhancing understanding of extremal problems for typically real and nonnegative sine polynomials.

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 $(0,π)$, W. Rogosinski and G. Szegö derived, among other things, exact bounds for $|b_2|$ via the Lukács 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 Fejér -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.

Theorems & Definitions (53)

• Theorem 3.1
• proof
• Corollary 3.1.1
• proof
• Theorem 3.2
• proof
• Corollary 3.2.1
• proof
• Theorem 3.3
• proof