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Unformatted text preview: 12 . 26 . 4312 . 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 ππ 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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 Spring '06
 PANTANO
 Multivariable Calculus, Least Squares, Cos, Continuous function, piecewise continuous functions

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