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Lecture 8 Notes

# 5 f x 1 1 2 2 1 05 1 2 10 10 15 15 1 0 1x0 0x1 2 05 05

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Unformatted text preview: e on [- 1,1] 101 term approximaSon to square wave on [- 1,1] 1.5 1.5 1.5 1.0 1.0 0.5 -2 -1 -0.5 1.5 1.0 0.5 1 2 , -2 -1 -0.5 2 -2 -1 -0.5 -1.0 -1.0 -1.5 1.5 1.0 1.0 0.5 -1 , 0.5 1.5 1.0 2 -1.5 1.5 1 -1.0 -1.5 -2 1.0 0.5 1 , -0.5 0.5 1 2 , -2 -1.0 -1 -0.5 f ( x) = ⇢ -1 1 2 , -2 -1 -0.5 1 2 -1.0 -1.0 -1.5 -1.5 1 0 1x<0 0x<1 2 -0.5 0.5 1 -1.0 -1.5 -2 f ( x) = ⇢ 1-1.5 0 1x<0 0x<1 1 14- 01- 31 Odd periodic extension from just 1 up to sum of 6 terms Error of 101- term approx 1.5 1.0 1.0 1.0 0.5 -3 0.5 -2 -1 1.0 0.5 1 2 3 , -3 -2 -1 0.5 1 2 3 , -3 -2 -1 1 -1.0 1.0 0.5 -3 -2 -1 1 2 3 , -3 -2 -1 0.5 1 2 3 , -3 -2 -1 -0.5 ⇢ -1.5 -0.5 -0.5 -1.0 -1.0 1 0 -1.0 -1.0 1x<0 0x<1 f ( x) = 100 term sum of odd periodic extension ⇢ 0x<1 1x<2 x 1 Even periodic extension 1 to 6 terms 1.0 1.2 1.2 1.0 1.0 0.8 0.8 0.6 0.5 1.2 1.0 0.8 0.6 , 0.6 , 0.4 -4 -1 1 0.4 0.2 0.2 -2 2 2 -4 4 -2 2 -4 4 -2 2 1.2 1.2 1.0 0.8 0.6 0.6 , 0.4 ⇢1.0 x f ( x) = 1 0x<1 1x<2 -2 4 0.8 0.6 , 0.4 0.2 -4 2 1.0 0.8 4 1.2 1.0 -0.5 , 0.4 0.2 -2 3 1.0 -0.5 f ( x) = 2 1.0 0.5 0.5 , -1.0 1.0 -0.5 3 -0.5 -1.0 -1.0 2 1 -0.5 -0.5 0.2 2 4 -4 f ( x) = -2 ⇢ 0.4 0.2 2 x 1 4 -4 -2 0x<1 1x<2 2 14- 01- 31 100 terms of even periodic extension 20 terms of even periodic extension 1.2 1.0 0.6 0.4 0.4 0.2 -2 0.8 0.6 f ( x) = 1.0 0.8 -4 1.2 0.2 ⇢ x 1 0 2 x < 1 1x<2 4 -4 -2 f ( x) = ⇢ x 1 0 2 x < 1 1x<2 4 3...
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