Gibbs Phenomenon: Estimating Oscillation Width

In summary, a student asked for help in estimating the width of oscillation in the Gibbs phenomenon when using Fourier series. Another student suggests using elementary calculus to find the maximum error, which does not decrease as more terms are added but the phenomenon decreases as one moves away from the discontinuity. They mention a book, "Introduction to the Theory of Fourier's Series and Integrals" by Horatio Scott Carslaw, which explains this concept in more detail.
  • #1
student85
138
0
Hi all,
I was asked something today at college. We're learning about Fourier series and we talked about the Gibbs phenomenon. The teacher asked us if we could possibly come up with a way of estimating the width of the oscillation in this phenomenon. I understand that increasing the number of terms in our series, doesn't decrease the phenomenon. Any help? Did I make myself clear? I'm looking for the width of the oscillation, an estimation.
Thanks in advance for any thoughts.
 
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  • #2
Use elementary calculus to find the maximum error. You will find that the maximum will not decreace as you add terms, but the phenomenon decreases as you move away from the discontinuity and do so faster with more terms.
say f(x+)-f(x-)~1
then let E(h) be the maximax error of approximations
E(h)~.09 (gibbs constant-1) (all h)
with
f(x*(h))=E(h)
but x*(h)->0
so the phenomonon is confined to smaller and smaller regions
see
Introduction to the Theory of Fourier's Series and Integrals By Horatio Scott Carslaw
pp268-273
on books.google.com
 
  • #3
in other words if the worst error is in the region between the maximum and minimum error as found by settting the derivative to 0 and solving.
 

What is Gibbs Phenomenon?

Gibbs Phenomenon is a mathematical phenomenon that occurs when approximating a discontinuous function using a finite series of continuous functions. It results in oscillations or overshoots near the discontinuity in the approximation.

How does Gibbs Phenomenon affect estimating oscillation width?

Gibbs Phenomenon can cause errors in estimating the width of oscillations because the oscillations near the discontinuity can artificially increase the estimated width.

Can Gibbs Phenomenon be avoided?

Gibbs Phenomenon is a fundamental property of mathematical approximation and cannot be completely avoided. However, it can be minimized by using more terms in the approximation or using special techniques such as the Lanczos sigma factor.

What is the significance of estimating oscillation width?

Estimating oscillation width is important in many fields such as signal processing, image processing, and physics. It can help in understanding the behavior of oscillatory functions and can be used to design filters and remove noise from signals.

Are there any applications of Gibbs Phenomenon in real-world problems?

Yes, Gibbs Phenomenon has been observed in various real-world problems such as signal processing, image reconstruction, and solving differential equations. Understanding and mitigating its effects is important in accurately solving these problems.

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