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I'm currently studying fourier seriers in Walter Rudin Principles of Mathematical Analysis, where the following defintion is made $$s_N(x)=s_N(f;x)= \sum _{-N}^N c_n e^{inx}$$ where $f$ is a function definied on $[ - \pi, \pi]$. What does $s_N(f;x)$ mean, or more generally, what changes if you have $g(x)$ and instead write $g(f;x)$? My only guess is that it represents that the domain of $x$ is governed by $f$, is this correct?

Thanks in advance

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  • $\begingroup$ Surely Rudin defines carefuly every symbol in his book. No need to guess. $\endgroup$ – uniquesolution Aug 13 '18 at 11:58
  • $\begingroup$ @uniquesolution I searched the "List of special symbols" but couldn't find anything about it. $\endgroup$ – Periodic Sqare well Aug 13 '18 at 12:02
  • $\begingroup$ Not every symbol is listed among the special symbols! Starting from the formula you quoted from the book, work your way backwards, carefully, page by page, until you reach the precise definition. Go on, try it. $\endgroup$ – uniquesolution Aug 13 '18 at 12:45
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$s_N(f;x)$ means the $N$-th partial sum of the Fourier series of the function $f$ evaluated at the point $x$.

  • $s$ is for sum
  • $N$ indicates the partial sum goes from $-N$ to $N$
  • $f$ is the function whose partial Furier sum we are evaluating
  • $x$ is the point where the partial sum is evaluated

If you want the partial Fourier sum of another function $g$ you would write $s_N(g;x)$.

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