Assume $\displaystyle f: [0, 2\pi] \rightarrow \mathbb{R}$ is Lipschitz continuous. Prove that exists constans $\displaystyle C>0$ that for every $\displaystyle k=1,2...$ there is:

$\displaystyle \int^{2\pi}_{0} f(x) \sin (kx) dx \leq \frac{C}{k}$.

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- Sep 3rd 2011, 10:08 AMCamille91Lipschitz continuous
Assume $\displaystyle f: [0, 2\pi] \rightarrow \mathbb{R}$ is Lipschitz continuous. Prove that exists constans $\displaystyle C>0$ that for every $\displaystyle k=1,2...$ there is:

$\displaystyle \int^{2\pi}_{0} f(x) \sin (kx) dx \leq \frac{C}{k}$.