Prove $\sin(kx)/x$

Question

I know that the infinite series

$$2\sum_{k=1}^{\infty} (-1)^{k+1} \frac{\sin(k x)}{k}=x \quad\quad \tag{1} $$

is the well known Fourier transform of $f(x)=x$. I would like to find upper bound for the partial sum of $(1)$, i.e.,

$$2\sum_{k=1}^{n} (-1)^{k+1} \frac{\sin(k x)}{k} <C$$

It would be particurarly nice if I could find an uper bound of the form

$$2\sum_{k=1}^{n} (-1)^{k+1} \frac{\sin(k x)}{k} <x+C$$

Answer 1

This may not be a complete answer, but I believe it provides some insight.

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I assume by "take $x$ to be on the interval $0<x<\pi$" in your comment you mean

$$f(x)=(x \bmod \pi)\tag{1}$$

which is approaximated by the Fourier series

$$f_n(x)=\frac{\pi }{2}-\lim\limits_{n\to\infty}\left(\sum\limits_{k=1}^n \frac{\sin (2 k x)}{k}\right)\tag{2},$$

but Gibbs phenomenon-Description indicates the following:

"As more Fourier series constituents or components are taken, the Fourier series shows the first overshoot in the oscillatory behavior around the jump point approaching ~ 9% of the (full) jump and this oscillation does not disappear but gets closer to the point so that the integral of the oscillation approaches to zero (i.e., zero energy in the oscillation)."

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The following plot illustrates $f_n(x)-f(x)$ where formula (2) above for $f_n(x)$ is valuated at $n\in\{1, 2, 4, 8, 16, 32\}$ and the horizontal reference line is at $y=0.09\, \pi$.