Given the ability to build any trigonometric polynomial of integer degrees:
$$T_d(\theta) = \sum_{n=-d}^d c_n e^{in\theta}$$
I wish to construct, or technically approximate:
$$e^{it\theta} \>\>\> \mbox{for $t\in [\frac{-1}{2},\frac{1}{2}$}]$$
I'm having trouble finding the best way to do this. When I use Wolfram Alpha to calculate the Fourier Series for $e^{it\theta}$ I get the following:
$$F_N(\theta) = \sum_{n=-N}^N \frac{(-1)^n sin(\pi t)}{\pi (t - n)}e^{in\theta}$$
- I'm not sure how it calculates this since $e^{it\theta}$ wouldn't generally be $2\pi$-periodic.
- Furthermore, here it is mentioned that the Fourier coefficients of an analytic function must decay exponentially; however, in this case the coefficients decay linearly even though $e^{it\theta}$ is analytic which really confuses me.
My main question is, what's the best/most efficient way of building an $\epsilon$-approximation of $e^{it\theta}$ using $T_d(\theta)$? I would also want to know the asymptotic complexity of such an approximation in $d$. As in, would I need $O(\frac{1}{\epsilon})$ terms? $O(\frac{1}{\epsilon^2})$ terms? $O(\log(\frac{1}{\epsilon}))$ terms?
This is a part of some algorithmic research I'm doing that could potentially have a significant impact. Unfortunately however, I don't have a formal background in complex/Fourier analysis so any insight into achieving this or even directions to look into is greatly appreciated! Also, any insight into (1) or (2) would also be very helpful!