On One Inequality of Different Metrics for Trigonometric Polynomials

Abstract

We study the sharp inequality between the uniform norm and \(L^p(0,\pi/2)\)-norm of polynomials in the system \(\mathscr{C}=\{\cos (2k+1)x\}_{k=0}^\infty\) 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 \(n\to\infty\) and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.