TAMU MATH 311 - Fourier-series (6 pages)

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Fourier-series

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Fourier-series

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Pages:
6
School:
Texas A&M University
Course:
Math 311 - Top In Applied Math I
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Math 311 handout on Fourier series Michael Anshelevich March 14 2008 f x x Fourier series SN x 2 PN 1 k k 1 k sin kx 3 2 1 0 3 2 1 x 0 1 2 3 1 2 3 Using 4 8 16 terms on Approximates well except at the endpoints values 0 instead of 1 4 PN 1 f x x Fourier series UN x 1 2 k 0 2k 1 2 cos 2k 1 x 3 0 2 5 2 0 1 5 1 0 0 5 0 0 3 2 1 0 1 2 3 x With 0 1 terms Why are so few terms enough 2 f x 1 1 x 0 Fourier series 0 x N 4 X 1 TN x sin 2k 1 x k 0 2k 1 1 0 0 5 0 0 3 2 0 1 1 2 3 x 0 5 1 0 3 Approximation on the whole real line with 50 terms Note the end point jumps on the first but not the second graph and periodicity Also note the Gibbs phenomenon 12 5 1 0 05 10 105 105 5 10 105 10 10 0 0 5 5 7 5 x 0 5 0 0 0 5 5 10 0 0 10 2 5 x 0 0 0 5 10 x 4 Gibbs phenomenon does not disappear for better approximations for 50 vs 100 terms 1 1 1 0 0 9 0 8 0 7 0 6 0 5 0 0 0 05 0 1 0 15 0 2 x Pointwise but not uniform convergence 5

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