The Fourier Convergence Theorem
Suppose \(f: \mathbb{R} \to \mathbb{R}\) is \(2\pi\)-periodic and piecewise smooth. Then, for each \(\theta \in \mathbb{R}\), we have \[ \lim_{N\to\infty} S_N^f(\theta) = \frac{\lim\limits_{\psi \to \theta^+} f(\psi) + \lim\limits_{\psi \to \theta^-} f(\psi)}{2}, \] where the limits are the right- and left-hand limits of \(f\) at \(\theta\). In particular, if \(f\) is continuous at \(\theta\), then \[ \lim_{N\to\infty} S_N^f(\theta) = f(\theta). \]
For simplicity, we will only prove the theorem in the case that \(f\) is actually smooth.
We define an auxiliary function \(g\) by setting \[ g(\phi) = \frac{f(\theta + \phi) - f(\theta)}{e^{i\phi}-1} \] for all \(\phi\neq 0\). Note that \(g(0)\) is not defined.
However, by l’Hôpital’s rule, we have that \[ \lim_{\phi \to 0} g(\phi) = \lim_{\phi\to 0} \frac{f'(\theta + \phi)}{i e^{i\phi}} = -i f'(\theta). \]
This is where smoothness of \(f\) is used! Hence, we can extend \(g\) to a smooth function on all of \(\mathbb{R}\) by defining \(g(0) = -i f'(\theta)\).
Notice that \(g\) is now \(2\pi\)-periodic and smooth. Hence, it is integrable and it has Fourier coefficients \[ C_N = \frac{1}{2\pi} \int_{-\pi}^\pi g(\phi) e^{-i N \phi} \, d\phi. \]
By the the Riemann-Lebesgue lemma, we have \[ \lim_{N\to \infty} \left( C_{-(N+1)} - C_{-N} \right) = 0, \] since both \(C_{-(N+1)}\) and \(C_{-N}\) tend to zero as \(N\to \infty\).
However, we have \[ C_{-(N+1)} - C_{N} = \frac{1}{2\pi} \int_{-\pi}^\pi g(\phi) \left( e^{i(N+1)\phi} - e^{-i N \phi} \right) \, d\phi. \]
Recall our expression for the \(N\)-th Dirichlet kernel and the \(N\)-th partial sum of the Fourier series: \[ D_N(\phi) = \frac{1}{2\pi} \frac{e^{i(N+1)\phi} - e^{-i N \phi}}{e^{i\phi}-1}, \quad S_N^f(\theta) = \frac{1}{2\pi} \int_{-\pi}^\pi f(\theta + \phi) D_N(\phi) \, d\phi. \]
Thus: \[ \begin{align*} C_{-(N+1)} - C_{N} &= \int_{-\pi}^\pi \left[f(\theta + \phi) - f(\theta)\right] D_N(\phi) \, d\phi \\ &= \int_{-\pi}^\pi f(\theta + \phi) D_N(\phi) \, d\phi - f(\theta) \int_{-\pi}^\pi D_N(\phi) \, d\phi \\ &= S_N^f(\theta) - f(\theta). \end{align*} \]
But we know that \(\lim_{N\to \infty} \left( C_{-(N+1)} - C_{-N} \right) = 0\), so we conclude that \[ \lim_{N\to \infty} S_N^f(\theta) = f(\theta). \]
Recall the \(2\pi\)-periodic extension of the function \(f(\theta) = \theta\) for \(\theta \in [-\pi, \pi]\). We discovered that its Fourier series is given by \[ 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin{n\theta}. \]
Recall the \(2\pi\)-periodic extension of the function \(f(\theta) = |\theta|\) for \(\theta \in [-\pi, \pi]\). We discovered that its Fourier series is given by \[ \frac{\pi}{2} - \frac{4}{\pi} \sum_{n=1}^\infty \frac{\cos{\left[(2n-1)\theta\right]}}{(2n-1)^2}. \]