PS5 - f ( x ) = 7 . 6 cos 2 x ? What multiple of sin x is...

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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 h U, V i = ( u 1 u 2 ) ± 2 1 1 3 ²± v 1 v 2 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 h x , y i = X 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 i i and g = X i i , with normalized orthogonal basis vectors { i } and the L 2 [ a, b ] inner product, we have the generalized Parsevals equality: Z b a f ( x ) g ( x ) dx = X n =1 n n Version 1.1...
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