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

For the class of sine polynomials $b_1\sin t+b_2\sin 2t+\mathrm{.}\mathrm{.}\mathrm{.}+b_N\sin Nt,(b_N≠0),$ which are nonnegative on $(0,\pi )$, W. Rogosinski and G. Szegö derived, among other things, exact bounds for $\mathrm{\mid }{b}_2\mathrm{\mid }$ 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.