On One Inequality of Different Metrics for Trigonometric Polynomials

Abstract

We study the sharp inequality between the uniform norm and Lp(0,π∕2)-norm of polynomials in the system = {cos(2k + 1)x}k=0∞ of cosines with odd harmonics. We investigate the limit behavior of the best constant in this inequality with respect to the order n of polynomials as n → ∞ and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.