----- - Let $f$ be a real-valued function [[function on the (unit) circle|on the circle]]. > [!proposition] Proposition. ([[conjugate symmetry property of Fourier series]]) > We have $\hat{\overline{f}}(n)=\overline{\hat{f}}(-n)$ for all $n \in \zz$, where $\hat{f}(n)$ denotes the $n^{th}$ [[Fourier series|Fourier coefficient]] of $f$. > [!proof]- Proof. ([[conjugate symmetry property of Fourier series]]) > Write $\begin{align} \hat{\overline{f}}(n) & =\frac{1}{2\pi}\int_{-\pi}^{\pi} \overline{f(\theta)}e ^{-in \theta} \, d\theta \\= & \frac{1}{2\pi} \int_{-\pi}^{\pi}\overline{f(\theta)e^{in \theta}}\, d\theta \\ = & \overline{ \frac{1}{2\pi} \int_{-\pi}^{\pi}f(\theta)e^{in \theta}\, d\theta} \\ = & \overline{\hat{f(-n)}}. \qedin \end{align}$ ----- #### ---- #### 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 > ```