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