## Mathematica plots

Bell, Jordan. “Estimates for the Norms of Products of Sines and Cosines.” Journal of Mathematical Analysis and Applications 405, no. 2 (2013): 530–45. https://doi.org/10.1016/j.jmaa.2013.04.010.

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$P_n(\theta) = \prod_{k=1}^n (1-e^{ik\theta})$ $P_n(\theta)=(-2i)^n e^{\frac{iN\theta}{2}} \prod_{k=1}^n \sin\bigg(\frac{k\theta}{2} \bigg), \qquad N=\frac{n(n+1)}{2}$ $‖f‖_{L^1} = \frac{1}{2\pi} \int_0^{2\pi} |f(\theta)| d\theta$ $K = \log 2 + \max_{0 < w < 1} \Bigg( \frac{1}{w} \int_0^w \log \sin(\pi t) dt \Bigg)$ $‖f‖_{L^2} = \left(\frac{1}{2\pi} \int_0^{2\pi} |f(\theta)|^2 d\theta\right)^{\frac{1}{2}}$ $Q_n(\theta) = \prod_{k=1}^n (1+e^{ik\theta})$ $Q_n(\theta)=2^n e^{\frac{iN\theta}{2}} \prod_{k=1}^n \cos\big(\frac{k\theta}{2} \big), \quad N=\frac{n(n+1)}{2}$ $\widehat{f}(k) = \int_0^{2\pi} f(\theta) e^{-ik\theta} d\theta,\quad k \in \mathbb{Z}$ $‖\widehat{f}‖_{\ell^1} = \sum_{k \in \mathbb{Z}} |\widehat{f}(k)|$ $‖\widehat{f}‖_{\ell^3} = \left( \sum_{k \in \mathbb{Z}} |\widehat{f}(k)|^3 \right)^{\frac{1}{3}}$

## Desmos plots

product of sin(kx)

product of cos(kx)