Gibbs Phenomenon: Estimating Oscillation Width

[student85]

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.

 

[lurflurf]

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

 

[lurflurf]

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.