# HW4 - II 9 For a fixed a>0 compute the following sum...

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Homework #4 Due: Tuesday, March 3, 2009 Note : Use of Matlab (or any other software) is not permitted. I. Compute the Fourier transform of the following functions (1-8): 1. = otherwise x x f , 0 2 1 , 1 ) ( 2. < < < + < = 3 , 0 3 1 , 2 / ) 3 ( 1 1 , 1 1 3 , 2 / ) 3 ( 3 , 0 ) ( x x x x x x x x f 3. | | ) ( x a e x f = , for some a>0. 4. 2 2 1 ) ( a x x f + = , for some a>0. (Hint: Use synthesis formula and exercise 3) 5. ( ) = otherwise x x x f , 0 2 1 , 2 sin ) ( π 6. () ax e x f x 2 cos ) ( | | = , for some a>0. 7. () 2 2 2 cos ) ( a x x x f + = , for some a>0. 8. c bx ax x f + + = 2 1 ) ( , for some real a,b,c so that a>0 and b 2 -4ac<0. (Hint: Complete the square.)

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Unformatted text preview: II. 9. For a fixed a >0, compute the following sum using the Poisson summation formula: ∑ ∞ −∞ = + − n a n x 2 2 ) ( 1 10. Let f be a piecewise smooth function on R that vanishes outside some finite interval. Show that 1 ) ( = − ∑ ∞ −∞ = m m x f (i.e. we can partition unity with the 1-translates of f) if and only if ⎩ ⎨ ⎧ ± ± = = = ,... 2 , 1 k k 1 ) ( for for k F Total : 10 pts (1 point each)...
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HW4 - II 9 For a fixed a>0 compute the following sum...

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