Meaning om notation $g(f;x)$

Question

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

Answer 1

$s_N(f;x)$ 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 $s_N(g;x)$.