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 ${\prod }_{k=1}^n\sin \left(\frac{k\theta }{2}\right)$ and ${\prod }_{k=1}^n\cos \left(\frac{k\theta }{2}\right)$ 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 $Q_n$.

arXiv comment18 pages