Estimates for the norms of products of sines and cosines

Abstract

In this paper we prove asymptotic formulas for the $L^p$ norms of $P_n(\theta)=\prod_{k=1}^n (1-e^{ik\theta})$ and $Q_n(\theta)=\prod_{k=1}^n (1+e^{ik\theta})$. These products can be expressed using $\prod_{k=1}^n \sin\Big(\frac{k\theta}{2}\Big)$ and $\prod_{k=1}^n \cos\Big(\frac{k\theta}{2}\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of $Q_n$.