- #1

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I've got some problems on the following:

Let [tex]f:\mathbb{R} \to \mathbb{R}[/tex] be a twice differentiable function with [tex]\lim_{x \to \infty} \frac{|f(x)|}{|x^2|}=0[/tex] and [tex]\int_{-\infty}^{\infty} |f''(x)| dx[/tex] is bounded.

Let

[tex]F(y) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi ixy} dx[/tex]

be the Fourier transform of [tex]f[/tex].

Then:

[tex]\sum_{n = -\infty}^{\infty} f(n) = \sum_{m = -\infty}^{\infty} F(m)[/tex]

holds.

For sure I can insert the Fourier transform of [tex]f[/tex] 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 :(