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Limit of a trigonometric integral

Let A \subseteq \mathbb{R} be a measurable of finite length set. Evaluate the limit:

    \[\ell = \lim_{\lambda \rightarrow +\infty} \int \limits_{A} | \sin \lambda x | \, {\rm d}x\]

Solution

Let us begin with the Fourier series of |\sin x | which is of the form:

    \[|\sin x | = \frac{2}{\pi} + \sum_{n=1}^{\infty} a_n \cos nx\]

Hence

(1)   \begin{equation*} |\sin \lambda x | = \frac{2}{\pi} + \sum_{n=1}^{\infty} a_n \cos n \lambda x  \end{equation*}

Integrating ( 1 ) we get that

    \begin{align*} \ell &= \lim_{\lambda \rightarrow +\infty} \int \limits_{A} \left | \sin \lambda x \right | \, {\rm d}x \\ &=\lim_{\lambda \rightarrow +\infty} \int \limits_{A} \left ( \frac{2}{\pi} + \sum_{n=1}^{\infty} a_n \cos n \lambda x \right ) \, {\rm d}x \\ &= \lim_{\lambda \rightarrow +\infty} \int \limits_{A} \frac{2}{\pi} + \lim_{\lambda \rightarrow +\infty} \sum_{n=1}^{\infty} a_n \int \limits_{A} \cos n \lambda x \, {\rm d}x \\ &= \frac{2 \left | A \right |}{\pi} \end{align*}

since it follows from Riemann – Lebesgue lemma that

    \[\lim_{\lambda \rightarrow +\infty} \int \limits_{A} \cos n \lambda x \, {\rm d}x = 0\]

 

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