soln10 - R. Balan Homework #10 Solutions MATH 464 I . (a)...

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Unformatted text preview: R. Balan Homework #10 Solutions MATH 464 I . (a) Poisson's summation formula implies: f ( x ) = X m =- e- ( x- mp ) 2 = 1 p X k =- e 2 ikx/p F ( k/p ) where F ( s ) = F ( e- x 2 )( s ) = e- s 2 Thus: f ( x ) = 1 p X k =- e 2 ikx/p e- k 2 /p 2 which is the Fourier series expansion of f ( x ) . (b) Poisson's sum formula implies: f ( x ) = X m =- e- a | x- mp | = 1 p X k =- e 2 ikx/p F ( k/p ) where F ( s ) = F ( e- a | x | )( s ) = 2 a a 2 + 4 2 s 2 Thus f ( x ) = 1 p X k =- e 2 ikx/p 2 ap 2 a 2 p 2 + 4 2 k 2 which is the Fourier series expansion of f ( x ) . II . Recall c n = R 1 e- 2 inx f ( x ) dx . Thus (a) Assume f ( x ) = f ( x ) . Thus: c n = Z 1 e 2 inx f ( x ) dx = Z 1 e 2 inx f ( x ) dx = c- n . (b) Assume f ( x ) =- f ( x ) . Thus: c n = Z 1 e 2 inx f ( x ) dx =- Z 1 e 2 inx f ( x ) dx =- c- n . (c) Assume f ( x + 1 /m ) = f ( x ) . Thus: c n = m- 1 X k =0 Z ( k +1) /m k/m e- 2 inx f ( x ) dx = m- 1 X...
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This note was uploaded on 05/05/2010 for the course MATH 464 taught by Professor Staff during the Spring '08 term at Maryland.

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soln10 - R. Balan Homework #10 Solutions MATH 464 I . (a)...

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