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Let's say I'm approximating $f(x) = e^{itx}$ for $t\in [0,\frac{1}{2}]$, using its truncated Fourier series $F_n(x)$.

I want to know, for an $\epsilon$-approximation of $f$, what would be the asymptotic complexity of $n$? As in, would I need $O(\frac{1}{\epsilon})$ terms? $O(\frac{1}{\epsilon^2})$ terms? $O(\log(\frac{1}{\epsilon}))$ terms? I'm not sure how to approach this problem, I've looked at some resources online but I'm still uncertain.

I believe the truncated Fourier expansion for $f$ is:

$$F_n(x) = \sum_{j=-n}^n \frac{(-1)^j sin(\pi t)}{\pi (t - j)}e^{ijx}$$

Any insight is greatly appreciated!

Dani007
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  • How do you measure the error? $L^2$? – PhoemueX Aug 04 '22 at 20:58
  • Is $[0,1/2]$ the a fundamental domain of the Fourier series? Or do you envision a larger period, for instance $2\pi$ with domain $[-\pi,\pi]$, and want to control the convergence on the smaller interval $[0,1/2]$? Note that you get a jump discontinuity at the wrap-around for almost every value of $t$. – Lutz Lehmann Aug 05 '22 at 15:59

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