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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 $x_c$)?

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  • $\begingroup$ 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

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