Math 311 handout on Fourier seriesMichael AnshelevichMarch 14, 2008f(x) = x. Fourier series SN(x) = −2PNk=1(−1)kksin(kx).−20−2223−30−11−1−3 31xUsing 4, 8, 16 terms on [−π, π]. Approximates well, except at the endpoints:values 0 instead of ±π.1f(x) = |x|. Fourier series UN(x) =12π −4πPNk=01(2k+1)2cos((2k + 1)x).1.50.53x3.02.012.521.00.00−1−2−3With 0, 1 terms. Why are so few terms enough?2f(x) =−1, −π ≤ x < 0,1, 0 ≤ x ≤ π. Fourier seriesTN(x) =4πNXk=012k + 1sin((2k + 1)x).−2−3x3−1−0.50.01.00 10.5−1.023Approximation 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 Gibbsphenomenon.x−5100−50510−10−1052.512.510.0100.05−57.50−10x5.0−1.01.0−0.5−5x100.05−1000.54Gibbs phenomenon does not disappear for better approximations: for 50 vs.100 terms0.050.61.11.00.9x0.50.150.0 0.20.80.70.1Pointwise but not uniform
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