Hi I found this in an Advanced Calculus book which does not go into the details of real analysis. Hence, I was wondering what would be an "elementary proof" of this result.

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Yes, I understand that this is the Poisson Summation Formula.

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- Jul 3rd 2009, 05:32 PMnonsingularSeeking for a Simple Proof
Hi I found this in an Advanced Calculus book which does not go into the details of real analysis. Hence, I was wondering what would be an "elementary proof" of this result.

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*Yes, I understand that this is the Poisson Summation Formula.* - Jul 4th 2009, 06:10 AMhalbard
Depends on what you mean by "elementary" I suppose...

Assume suitable conditions on the function , such as with which ensure convergence of the relevant series.

Let . This series converges uniformly to a function in and can be expanded in a uniformly convergent Fourier series:

where

.

Here is the Fourier transform of .

Thus .

To obtain your result use and change to in the integral.