The coefficient of each term in a Taylor series describes the series' nth derivative as aligning with that of the function $f(x)$ it aims to describe. What aspect of the function $g(x)$ a Fourier series aims to describe does its coefficient represent?
I understand that a Fourier series's coefficients represent its alignment with its sinusoidal component of "$n$-th frequency," which is analogous in one sense to how a Taylor series' coefficient describes the degree to which an nth-degree monomial is represented in a function. However, that's a different relationship to what I'm asking about and is the only thing I've found so far in my research. I've noticed that a Fourier coefficient is like an average over a circle, which I think is on the right track, but I can't figure out what property of the original function it represents.
Glossing over some non-trivial convergence details, both developments are computing the projection of a function over an (infinite dimension) vector space representing (most of the common) univariate functions.
In the case of the Fourier series, the basis used is something like $e^{inx}$. In the case of the Taylor expansion, the basis is $x^n$.
There are other differences but this is how I find it intuitive.