----- - Let $f$ be a function [[function on the (unit) circle|on the circle]]. > [!proposition] Proposition. ([[bounded function has bounded Fourier coefficients]]) > If $\ex B>0$ s.t. $|f(\theta)| \leq B$ for all $\theta$, then $|\hat{f}(n)| \leq B$ for all $n \in \zz$. > > [!proof]- > $\begin{align} |\hat{f}(n)|=&\frac{1}{2\pi} \left|\int _{-\pi} ^{\pi}f(\theta) e ^{-in \theta} \, d\theta \right| \\ \leq& \frac{1}{2\pi} \int _{-\pi} ^{\pi} | f(\theta) | \, d\theta , \ \text{(since $|e ^{-in \theta}|=1$)}\\ \leq& \frac{1}{2\pi} \int_{-\pi}^{\pi} B \, d \theta \\ =& B. \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```