Sunday, 22 February 2009

ca.analysis and odes - What does Gibbs phenomenon shows the nature of Fourier Series

Hi there,



I think there is a bit more to answering your question than considering just the strict L2 convergence however. The Gibbs phenomenon is important when considering the pointwise convergence of the partial sums of the fourier series. When fprime is continuous on a compact interval, you will get pointwise convergence of the partial sums SN so SN(x)tof(x) as Ntoinfty. (You only get uniform convergence if in addition the function f is compatible with the boundary conditions for your expansion but this is not your question anyway).



Now what happens when f is discontinuous? It turns out that SN(x)tofrac12[f(x)+f(x+)], the average of the left and right limits of the function. However, this is really only true for Ntoinfty. Otherwise for any finite N there is a small width of order 1/N around your discontinuous point x where your partial sums are uniformly bounded away from either value f(x+), f(x) by some fixed percentage (I recall 9% of the jump size or something but don't quote me on that). Check out the photos at:http://en.wikipedia.org/wiki/Gibbs_phenomenon
That little wiggle of the wave where the jump of f occurs stays uniformly bounded away from the value of the function but the width of this region (1/N) goes to zero as Ntoinfty so that technically you still get the full pointwise convergence.

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