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soln10

# 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- π ( x- mp ) 2 = 1 p ∞ X k =-∞ e 2 πikx/p F ( k/p ) where F ( s ) = F ( e- πx 2 )( s ) = e- πs 2 Thus: f ( x ) = 1 p ∞ X k =-∞ e 2 πikx/p e- πk 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 πikx/p F ( k/p ) where F ( s ) = F ( e- a | x | )( s ) = 2 a a 2 + 4 π 2 s 2 Thus f ( x ) = 1 p ∞ X k =-∞ e 2 πikx/p 2 ap 2 a 2 p 2 + 4 π 2 k 2 which is the Fourier series expansion of f ( x ) . II . Recall c n = R 1 e- 2 πinx f ( x ) dx . Thus (a) Assume f ( x ) = f ( x ) . Thus: c n = Z 1 e 2 πinx f ( x ) dx = Z 1 e 2 πinx f ( x ) dx = c- n . (b) Assume f ( x ) =- f ( x ) . Thus: c n = Z 1 e 2 πinx f ( x ) dx =- Z 1 e 2 πinx 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 πinx f ( x ) dx = m- 1 X...
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