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arXiv · 1210.7380

Estimates for the norms of products of sines and cosines

Jordan Bell
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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 $Q_n$.

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Jordan Bell. 2012-10-27. Estimates for the norms of products of sines and cosines. https://arxiv.org/abs/1210.7380

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