Gibbs Phenomenon: Investigating Fourier Series of a Discontinuity

In summary, the conversation discusses the Gibbs phenomenon and the behavior of a Fourier series of a region of discontinuity of a function near 0. They mention that since the function is odd, it is sufficient to look at its behavior at the interval [0, pi]. The last question asks for clarification on why this is the case.
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

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)
 
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  • #2
mathmari said:
Hey! :eek:

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

$\chi$ $\sigma$
 
  • #3
chisigma said:
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)
 

What is Gibbs Phenomenon?

Gibbs Phenomenon is a mathematical phenomenon that occurs when approximating a discontinuous function using a Fourier series. It is characterized by oscillations near the discontinuity, even as the number of terms in the Fourier series increases.

How does Gibbs Phenomenon affect Fourier series approximations?

Gibbs Phenomenon causes the Fourier series approximation to overshoot or undershoot the true value of a discontinuous function near the point of discontinuity. This results in oscillations that do not converge to the true value, even as the number of terms in the series increases.

What causes Gibbs Phenomenon?

Gibbs Phenomenon is caused by the fact that Fourier series use a finite number of terms to approximate a function, which is continuous and infinite. This leads to a mismatch between the finite number of terms used in the series and the infinite number of terms needed to accurately represent the discontinuity.

Can Gibbs Phenomenon be eliminated?

No, Gibbs Phenomenon cannot be completely eliminated. However, it can be minimized by using a larger number of terms in the Fourier series or by using alternative methods of approximation, such as the use of different types of basis functions.

What practical applications does Gibbs Phenomenon have?

Gibbs Phenomenon has practical applications in signal processing, image processing, and data compression. It is important to understand and account for Gibbs Phenomenon in these applications in order to obtain accurate approximations and avoid errors.

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