HW5 - integrable(that is ∞< ∞ ∞ − dx x g | and...

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Homework #5 Due: Tuesday, March 10, 2009 Note : Use of Matlab (or any other software) is not permitted. I. (Exercise 3.23) Let f be a suitably regular function on R with the Fourier transform F . What can you deduce about F if you know that: 1. 1 ) ( = dx x f 2. 1 ) ( = dx x xf 3. 0 ) ( ) 2 cos( = dx x f x π 4. 0 ) 0 ( ' = f 5. ) ( ) ( x f x f = II. (Exercise 3.28) Use your knowledge of Fourier analysis to find a function f that satisfies the given integral equation: 6. < < = x e du ux u f x 0 , ) 2 cos( ) ( 0 7. < < < < = x x du ux u f 1 if 0 1 0 if 1 ) 2 sin( ) ( 0 8. 2 ) ( ) ( x e du u x f u f = III. (Exercise 3.29) Let g be a piecewise smooth function on R which is absolutely
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Unformatted text preview: integrable (that is ∞ < ∫ ∞ ∞ − dx x g ) ( | ), and suppose that we wish to find such a function f that satisfies the differential equation: ∞ < < ∞ − = + − x x g x f x f , ) ( ) ( ) ( ' ' 9. Fourier transform the differential equation and thereby show that any suitably regular solution can be written in the form ∫ ∞ ∞ − − − = du u g e x f u x ) ( 2 1 ) ( | | 10. Find the function f and sketch its graph when ). ( ) ( x x g Π = Total : 10 pts (1 point each)...
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