Fouriertransformation, equation

[Tumbler]

Hi,

I've got some problems on the following:

Let $$f:\mathbb{R} \to \mathbb{R}$$ be a twice differentiable function with $$\lim_{x \to \infty} \frac{|f(x)|}{|x^2|}=0$$ and $$\int_{-\infty}^{\infty} |f''(x)| dx$$ is bounded.
Let
$$F(y) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi ixy} dx$$
be the Fourier transform of $$f$$.

Then:

$$\sum_{n = -\infty}^{\infty} f(n) = \sum_{m = -\infty}^{\infty} F(m)$$

holds.

For sure I can insert the Fourier transform of $$f$$ into the sum - but I don't see how to continue. Actually I assume it's a quite easy thing if one sees it... unfortunately I don't :(

 

[Dick]

You can write the one of the sums as a integral over a sum of delta functions. Change to a Fourier representation of the delta functions and rearrange it into the other sum. That's the 'engineering' approach. I'm not paying much attention to interchanging integration and summation, or to whether anything actually exists. Even more informally, you could move the summation inside the integral and note that the sum of the exponentials is the Fourier series representation of the 'Dirac comb'.