gibbs phenomenon

# gibbs phenomenon - AN INSTANCE OF GIBBS PHENOMENON KIAM...

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KIAM HEONG KWA The Gibbs phenomenon, named after the American physicist Josiah Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously diﬀerentiable periodic function behaves at a jump discontinuity: the partial sums of the Fourier series have large oscillations near the jump, which might increase the maxima of the partial sums above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a ﬁnite limit. Following [1], we illustrate the Gibbs phenomenon with the π -periodic function f ( x ) such that (1) f ( x ) = 1 2 for 0 < x < π, - 1 2 for - π < x < 0 . It Fourier series is F [ f ]( x ) = a 0 2 + X n =1 ( a n cos nx + b n sin nx ) (2) = X odd n =1 2 sin nx and the corresponding partial sums are F N [ f ]( x ) = a 0 2 + N X n =1 ( a n cos nx + b n sin nx ) (3) = N X odd n =1 2 sin nx Date : February 12, 2011. 1

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gibbs phenomenon - AN INSTANCE OF GIBBS PHENOMENON KIAM...

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