On 12 Jun 2015, at 14:31, Vadim Zavalishin 
<vadim.zavalis...@native-instruments.de> wrote:

> On one hand cos(omega0*t) is delta(omega-omega0)+delta(omega+omega0) in the 
> frequency domain (some constant coefficients possibly omitted). On the other 
> hand, its Taylor series expansion in time domain corresponds to an infinite 
> sum of derivatives of delta(omega). So an infinite sum delta^(n)(omega) 
> (which are zero everywhere except at the origin) must converge to 
> delta(omega-omega0)+delta(omega+omega0), correct? ;)

I’m not using any convergence properties in my argument. The space you’re 
arguing about is not complete, so convergence is not necessarily given.
My argument is simply that i*omega as an operator in frequency domain cannot 
have real eigenvalues. 

If the core of your argument is, that we could take \delta^{(n)} as a possible 
frame for constructing distributions that are eigenvectors of i*\omega, then 
that won’t work: i*\omega*\delta^{(n)}(\omega-\omega_0)  is again a Dirac 
distribution because it integrates like follows:

\int _{\omega_0-\eps}^{\omega_0+\eps} i \omega \delta^{(n)}(\omega-\omega_0) 
d\omega = (-1)^n i (d/d\omega0)^n \omega0 = 
\int_{\omega_0-\eps}^{\omega_0+\eps) (-1)^n i \delta(\omega-\omega_0) 
(d/d\omega0)^n \omega0 d\omega

therefore 

i*\omega*\delta^\{(n)}(\omega-\omega_0) = (-1)^n i (d/d\omega_0)^n \omega_0  
\delta(\omega-\omega_0)

Note that the right hand side vanishes for n>1, and only n=0 gives an 
eigenvector. So you cannot get an infinite series of Dirac derivatives, and 
even if you did, higher orders could not be combined to form eigenvectors, 
because they would all map to n=0. This argument holds for any distribution 
with single-point support. And as soon as you have non-single point support, 
you cannot get an eigenvector because of the non-degeneracy of the operator 
i*\omega.

I can’t really go into the details of the spectral theory of infinite 
dimensional representations of the Heisenberg algebra (which this is),as it’s 
one of the more complicated topics in mathematics and as far as I understand, 
it’s quite well understood. Questions about the existence of Fourier transforms 
map quite well to it. 

Andreas
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