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

2024/05/05 by Dmitrishin, Dmitriy, Stokolos, Alexander, Trebels, Walter · 1 citation
#42A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.02932

Abstract

For the class of sine polynomials b₁sin t+b₂sin2t+...+b_Nsin Nt, (b_N\not= 0), which are nonnegative on (0,π), W. Rogosinski and G. Szegö derived, among other things, exact bounds for |b₂| 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.

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• Extremal polynomials for the Rogosinski--Szegő estimates of the third coefficient of nonnegative sine polynomials

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