I want to approximate identity function $g(x) = x$ for $x \in [0,x_c]$ with $x_c<\pi/2$ by finite (sum) Fourier sine series $f(x)$. $f(x)<x_c$ is required for $x \in [0,\pi]$, $f(x)$ is assumed to be a $2\pi$-periodic function (that is, we can know everything about $f(x)$ by looking at its values for $x \in [-\pi,\pi]$), and approximation needs to be close within precision of $k$ fractional digits in binary or decimal for $x\in [0,x_c]$. $k$ is not fixed.

The question is, how many sine terms would be required to satisfy this, in function of $k$ (and $x_c$)?

  • $\begingroup$ Perhaps infinetely many, if you take Gibbs phenomenon into account ... en.wikipedia.org/wiki/Gibbs_phenomenon $\endgroup$ – denklo Jan 11 at 9:28
  • $\begingroup$ Is gibbq phenomen relevant? We have a continuouq function here. $\endgroup$ – Math_QED Jan 11 at 9:50
  • $\begingroup$ I don't know, like this is not about approximating g(x)=x exactly for $x \in [-\pi,\pi]$, which would run into Gibbs phenomenon problems. But it only asks to approximate precisely identity function for $x \in [0,x_c]$ and one is free to do whatever for other values of $x$ as long as $f(x)<x_c$, so that gives some freedom. $\endgroup$ – Sayako Nama Jan 11 at 9:58
  • $\begingroup$ @SayakoNama Ok, no worries then :) $\endgroup$ – denklo Jan 11 at 9:59
  • $\begingroup$ Well, one way to re-phrase the question is, is there a better way to approximate a sawtooth function if we can afford to allow for some deviations from the sawtooth function for $x \in [x_c,\pi]$... The Fourier series of sawtooth function tries to exactly become a sawtooth function for $x\in [-\pi,\pi]$, which is rendered impossible unless infinite sums are made. $\endgroup$ – Sayako Nama Jan 11 at 10:03

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.