Integrals of Products of Sines and Cosines

Mathematica plots
Desmos

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} (θ) = \prod_{k = 1}^{n} (1 - e^{i k θ})

P_{n} (θ) = (- 2 i)^{n} e^{\frac{i N θ}{2}} \prod_{k = 1}^{n} \sin (\frac{k θ}{2}), N = \frac{n (n + 1)}{2}

Plot of sine product for n=1 to 10 — $\prod_{k = 1}^{10} 2 | \sin (k θ) |$ for $0 \leq θ \leq \frac{π}{2}$

‖ f ‖_{L^{1}} = \frac{1}{2 π} \int_{0}^{2 π} | f (θ) | d θ

K = \log 2 + max_{0 < w < 1} (\frac{1}{w} \int_{0}^{w} \log \sin (π t) d t)

Plot of sine product L1 norms for n=1 to 400 — $\frac{‖ P_{n} ‖_{L^{1}}}{e^{n K} n^{- 1}}$ for $n = 1, \dots, 400$

‖ f ‖_{L^{2}} = {(\frac{1}{2 π} \int_{0}^{2 π} | f (θ) |^{2} d θ)}^{\frac{1}{2}}

Plot of sine product L2 norms for n=1 to 400 — $\frac{‖ P_{n} ‖_{L^{2}}}{e^{n K} n^{- 1 / 4}}$ for $n = 1, \dots, 400$

Q_{n} (θ) = \prod_{k = 1}^{n} (1 + e^{i k θ})

Q_{n} (θ) = 2^{n} e^{\frac{i N θ}{2}} \prod_{k = 1}^{n} \cos (\frac{k θ}{2}), N = \frac{n (n + 1)}{2}

Plot of cosine product for n=1 to 10 — $\prod_{k = 1}^{10} 2 | \cos (k θ) |$ for $0 \leq θ \leq \frac{π}{2}$

\hat{f} (k) = \int_{0}^{2 π} f (θ) e^{- i k θ} d θ, k \in Z

‖ \hat{f} ‖_{ℓ^{1}} = \sum_{k \in Z} | \hat{f} (k) |

Plot of l1 norms of Fourier coefficients of Pn for n=1 to 500 — $\frac{‖ {\hat{P}}_{n} ‖_{ℓ^{1}}}{e^{K n} n^{1 / 2}}$ for $n = 1, \dots, 500$

‖ \hat{f} ‖_{ℓ^{3}} = {(\sum_{k \in Z} | \hat{f} (k) |^{3})}^{\frac{1}{3}}

Plot of l3 norms of Fourier coefficients of Qn for n=1 to 400 — $\frac{‖ {\hat{Q}}_{n} ‖_{ℓ^{3}}}{2^{n} n^{- 1}}$ for $n = 1, \dots, 400$

Desmos plots

product of sin(kx)

product of cos(kx)