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M294F09WS_LeastSquares

# M294F09WS_LeastSquares - 12 26 431-2 899 2 Consider the...

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Math 2940 Worksheet: Ch. 5 Orthogonality and Least Squares October 22, 2009 1. Let S ( t ) be the number of daylight hours on the t th day of the year 2008 in Rome, Italy. We are given the following data for S ( t ): Day t S(t) February 1 32 10 March 17 77 12 April 30 121 14 May 31 152 15 We wish to fit a trigonometric function of the form f ( t ) = a + b sin 2 π 366 t + c cos 2 π 366 t to these data. Find the best approximation of this form, using least squares. (Only set up your matrices and the equation, do not actually perform the calculations). How many daylight hours does your model predict for the longest day of the year 2008? (The actual value is 15 hours, 13 minutes, 39 seconds. Again, just set up the problem) Use the fact that a b c *

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Unformatted text preview: 12 . 26 . 431-2 . 899 . 2. Consider the space, X , of all piecewise continuous functions on the interval [-π,π ] (having at most ﬁnitely many discontinuities) with the inner product deﬁned as < f,g > = 1 π Z π-π f ( t ) g ( t ) dt. • Compute || 1 || , || sin( t ) || , || cos( t ) || , || sin(2 t ) || , and || cos(2 t ) || . • Let T 2 be the set of all functions of the form f ( t ) = a + b 1 sin( t )+ c 1 cos( t )+ b 2 sin (2 t )+ c 2 cos (2 t ). Show T 2 is a subspace of the space X . • Find an orthonormal basis for T 2 . (Check it) • Compute Proj T 2 ( f ( t )) where f ( t ) = ± ,-π ≤ t < 1 , ≤ t ≤ π ....
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