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Unformatted text preview: y with the trigonometric polynomial
K K bk sin(kt). ak cos(kt) + f ( t) = k=1 k=0 We consider two approximations: one that minimizes the L2 -norm of the error, deﬁned as
π f −y 2 = −π (f (t) − y (t))2 dt 1/2 , and one that minimizes the L1 -norm of the error, deﬁned as
π f −y 1 = −π |f (t) − y (t)| dt. The L2 approximation is of course given by the (truncated) Fourier expansion of y .
To ﬁnd an L1 approximation, we discretize t at 2N points,
ti = −π + iπ/N, i = 1, . . . , 2N, and approximate the L1 norm as
2N f −y 1 ≈ (π/N ) i=1 | f ( ti ) − y ( ti ) | . (A standard rule of thumb is to take N at least 10 times larger than K .) The L1 approximation (or
really, an approximation of the L1 approximation) can now be found using linear programming.
We consider a speciﬁc case, where y is a 2π -periodic square-wave, deﬁned for −π ≤ t ≤ π as
y ( t) = 1 |t| ≤ π/2
0 otherwise. (The graph of y over a few cycles explains the name ‘square-wave’.)
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
- Fall '13
- The Aeneid