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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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 Spring '08
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 Fourier Series, CN, dx

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