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About LinkBacksHi 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
=\frac1{2\pi}\int_0^{2\pi}g(x)\mathrm e^{-\mathrm inx}\mathrm dx=\frac1{2\pi}\sum_{k=-\infty}^\infty\int_0^{2\pi}f(x+2k\pi)\mathrm e^{-\mathrm inx}\mathrm dx)
.
Hereis the Fourier transform of
Thus.
To obtain your result useand change
to
in the integral.
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