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# PS5 - f x = 7 6 cos 2 x What multiple of sin x is closest...

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MATH 580: Problem Set 5 due Tuesday, October 19, 2004 1. Given two arbitrary vectors U = ( u 1 , u 2 ) and V = ( v 1 , v 2 ) from a linear vector space S , show that U, V = ( u 1 u 2 ) 2 1 1 3 v 1 v 2 defines a valid inner product on S . Note that you will need to assume ei- ther that u i , v i R , or that in the definition above, ( u 1 u 2 ) ( ¯ u 1 ¯ u 2 ) . 2. (Keener 2.1.1) - you can skip this one, it was on PS4! 3. Keener 2.1.5 (p.94) 4. For the real sequence space 2 , show explicitly that x , y = i =1 x i y i is a valid inner product. 5. What constant function is ‘closest’ in the least squares sense to
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Unformatted text preview: f ( x ) = 7 . 6 cos 2 x ? What multiple of sin x is closest to f = cos 3 x ? 6. Keener 2.2.4 (p.95) 7. Assuming that the operations of summation and integration can be interchanged, show that for f = X α i ϕ i and g = X β i ϕ i , with normalized orthogonal basis vectors { ϕ i } and the L 2 [ a, b ] inner product, we have the generalized Parseval’s equality: Z b a f ( x ) g ( x ) dx = ∞ X n =1 α n β n Version 1.1...
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