2305 Convergence of Fourier Series

The period $2L$ function $f(t)$ is called piecewise smooth if there are a only finite number of points $0 \leq t_1 < t_2 < \ldots < t_n ≤ 2L$ where $f(t)$ is not differentiable, and if at each of these points the left and right-hand limits $\lim\limits_{t \to t_i^+} f'(t)$ and $\lim\limits_{t \to t_i^-} f'(t)$ exist (although they might not be equal).

Recall that when we first introduced Fourier series we wrote $$\begin{aligned} f(t)&\sim\frac{a_0}{2}+a_1\cos t+a_2\cos 2t+a_3\cos 3t + \ldots+b_1\sin t+b_2\sin 2t+b_3\sin 3t+\ldots\\ &=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos nt+b_n\sin nt\tag{1} \end{aligned}$$ where we used '∼' instead of an equal sign. The following theorem shows that our subsequent use of an equal sign, while not technically correct, is close enough to be warranted.

Theorem: If $f(t)$ is piecewise smooth and periodic then the Fourier series for $f$ 1. converges to $f(t)$ at values of $t$ where $f$ is continuous 2. converges to the average of $f(t^-)$ and $f(t+)$ where it has a jump discontinuity.

Example. Square wave. No matter what the endpoint behavior of $f(t)$ the Fourier series converges to:

Example. Continuous sawtooth: Fourier series converges to $f(t)$.