Estimates for the norms of products of sines and cosines

TL;DR

This paper derives asymptotic formulas for the $L^p$ norms of specific products of sines and cosines, providing insights into their maximum values and Fourier coefficients, which are relevant in harmonic analysis.

Contribution

It introduces new asymptotic formulas for the norms and maximum Fourier coefficients of products of sines and cosines, advancing understanding in harmonic analysis.

Findings

Asymptotic formulas for $L^p$ norms of $P_n$ and $Q_n$

Estimate for $P_n$ near its maximum point

Asymptotic formula for maximum Fourier coefficients of $Q_n"

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 $Q_n$.