Gibbs Phenomenon: Investigating Fourier Series of a Discontinuity

[mathmari]

Hey!

I am looking at the Gibbs phenomenon.
We want to look at the behaviour of a Fourier series of a region of discontinuity of $f$, especially we are looking at the function $f(x)=sgn(x), x \in [-\pi, \pi]$ near $0$.Since $f$ is odd, it suffices to look its behaviour at $[0, \pi]$.

Why does the last sentence stand?? (Wondering)

 

[chisigma]

Hey!

I am looking at the Gibbs phenomenon.
We want to look at the behaviour of a Fourier series of a region of discontinuity of $f$, especially we are looking at the function $f(x)=sgn(x), x \in [-\pi, \pi]$ near $0$.Since $f$ is odd, it suffices to look its behaviour at $[0, \pi]$.

Why does the last sentence stand?? (Wondering)

A function is 'odd' if is $\displaystyle f(x) = - f(- x)$, so that is $f(0)=0$. Regarding the Gibbs phenomenon see...

Gibbs Phenomenon -- from Wolfram MathWorld

Kind regards

$\chi$ $\sigma$

 

[mathmari]

A function is 'odd' if is $\displaystyle f(x) = - f(- x)$, so that is $f(0)=0$. Regarding the Gibbs phenomenon see...

Gibbs Phenomenon -- from Wolfram MathWorld

Kind regards

$\chi$ $\sigma$

And why is it sufficient to look at the half of the interval, $[0, \pi]$ ? (Wondering)