On the Partial Sums of a Fourier Series

We give sharp lower estimates for the partial sums of the Fourier series\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sin x+\frac{\cos2x}{2}+\frac{\sin 3x}{3}+\frac{\cos 4x}{4}+\cdots,$\end{document} with both an even and odd number of terms. Our results are obtained through a monotonicity property of their local minima. In particular, we prove that all partial sums are positive on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0,\pi)$\end{document}.