Which types of functions admit Taylor polynomials in the Fourier domain?
Ok, this may sound cryptic. Therefore I will try to explain it more in depth.
We want to approximate function $$x\to f(x) \text{ around } x_0 \to f(x_0)$$
We can look at normal Taylor polynomials first:
$$p_N(x) = \sum_{k=0}^{N}\frac{f^{(k)}(x_0)}{k!} (x-x_0)^k$$
Here all is fun and games and we see that we have well defined coefficients $$c_k=\frac{f^{(k)}(x_0)}{k!}$$ which probably are derived in your first Uni course on calculus.
But what if we instead use trigonometric polynomials?
$$p_N(x) = \sum_{k=0}^{N}a_k \sin(x-x_0)^k+ b_k\cos(x-x_0)^k$$
Can we derive which $a_k$, $b_k$ would do a best fit "close to" $x_0$?
Also, in what sense would this expansion be a "best fit" (for given number of terms $N+1$)?