> [!NOTE] Lemma > For any [[Continuous Real Function|continuous real function]] $\phi\in C^0([-\pi, \pi],\mathbb{C})$, the [[Fourier Transform|Fourier coefficients]] has [[Convergence|limit]] zero at infinity, that is $\lim_{ k \to \pm\infty } \hat{\phi}(k) = \lim_{ k \to \pm\infty } \frac{1}{2\pi} \int_{-\pi}^\pi \phi(x)e^{-ikx} \,dx = 0 $ or, equivalently, $\lim_{ k \to \pm \infty } \frac{1}{2\pi} \int_{-\pi}^\pi \phi(x)\cos(-kx) \,dx = 0 \quad \text{ and }\quad\lim_{ k \to \pm \infty } \frac{1}{2\pi} \int_{-\pi}^\pi \phi(x)\sin(-kx) \,dx = 0 .$ ###### Proof It follows from [[Extreme Value Theorem]] that $\phi$ attains a maximum on the compact interval $[-\pi,\pi]$ hence $\int_{-\pi}^\pi |\phi(x)|^2 dx<\infty$. It follows from [[Bessel's Inequality|Bessel's inequality]] that $\sum_{k\in \mathbb{Z}}|\hat{\phi}(k)|^2 \leq \int_{-\pi}^\pi |\phi(x)|^2 dx<\infty$. Since [[Terms of Convergent Series Tend to Zero]], we conclude that $\hat{\phi}(k)\to 0$ as $k\to \pm\infty$. $\blacksquare$