Exercise 11.17

Question

Suppose $E \subset (-\pi,\pi)$, $m(E) > 0$, $\delta > 0$. Use the Bessel
    inequality to prove that there are at most finitely many integers $n$ such
    that $\sin nx \ge \delta$ for all $x \in E$.

Amit Awasthi · Accepted Answer

\begin{proof}
      By the Bessel inequality (Theorem $8.12$) applied to the function $\chi_E$,
      we have
      \begin{equation*}
        \sum_{n=1}^\infty \left\lvert \int_E \sin nx\,dx \right\rvert \le m(E) <
        \infty.
      \end{equation*}
      Now if there are infinitely many $n$ such that $\sin nx \ge \delta$ for
      all $x \in E$, then the sum on the left would be infinite, contradicting
      that $m(E) < \infty$.
    \end{proof}

Exercise 11.17

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Exercise 11.17

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