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s4 - Problem Set 4 1(Problem of 3.6 Prove the dierentiation...

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Problem Set 4 1. (Problem of 3.6) Prove the diFerentiation and multiplication by x n formulas for the ±ourier transform. Answer It su²ces to illustrate the steps with the n = 1 cases. (a) As lim x →±∞ f ( x ) = 0, F [ f p ] = i -∞ f p ( x ) e - iωx dx = f ( x ) e - iωx v v v v -∞ + ( ) i -∞ f ( x ) e - iωx dx = ( ) F [ f ] (b) F [ xf ] = i -∞ e - iωx xf ( x ) dx = i d i -∞ e - iωx f ( x ) dx = i d F [ f ] 2. (Problem 11 of 3.6) ±ind the ±ourier transform of f ( x ) = e -| x | . Answer F [ f ] = i 0 e - (1+ ) x dx + i 0 -∞ e (1 - ) x dx = e - (1+ ) x - (1 + ) v v v v 0 + e - (1 - ) x - (1 - ) v v v v 0 = 2 1 + ω 2 3. (Problem 26 of 3.6) Prove Parseval’s theorem. Let f be real-valued, continuous and piecewise smooth on the real line. Assume that i -∞ | f ( x ) | dx and i -∞ f 2 ( x ) dx converge. Then i -∞ f 2 ( x ) dx = 1 2 π i -∞ | h f ( ω ) | 2 dω. Answer 1
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First, note that the complex conjugate of h f is ( h f ) * ( ω ) = pi -∞ fe - iωx dx P * = i -∞ f * e iωx dx = i -∞ iωx dx as f is real. Therefore i -∞ f 2 ( x ) dx = i -∞ f ( x ) pi -∞ 1 2 π h f ( ω ) e iωx P dx =
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