Given $S = \{e^{i\theta}: \theta \in \mathbb{R}\}$, $f: S \to\mathbb{C}$, and $g: S \to\mathbb{C}$ I'm trying to choose a definition to represent how well $f$ approximates $g$ on the unit circle. There are 2 definitions I've thought of:
- $\int_0^{2\pi}\vert\vert f(e^{i\theta}) - g(e^{i\theta})\vert\vert \>d\theta$
- An inner product definition like: $$\vert\vert f - g\vert\vert^2 = \><f-g,f-g> \>= \int_0^{2\pi}(f(e^{i\theta}) - g(e^{i\theta})) \overline{(f(e^{i\theta}) - g(e^{i\theta}))} \>d\theta$$
I'm not sure which to choose, and I'm not quite sure how to approach the "which is better?" question. Any insight into what each of these entities would represent intuitively, or which is a better measure of the quality of an approximation is appreciated! If it makes things easier, for my specific use case both $f$ and $g$ are polynomials.