On 12-Jun-15 12:54, Andreas Tell wrote:
I think it’s not hard to prove that there is no consistent
generalisation of the Fourier transform or regularisation method that
would allow plain exponentials. Take a look at the representation of
the time derivative operator in both time domain, d/dt, and frequency
domain, i*omega. The one-dimensional eigensubspaces of i*omega are
spanned by the eigenvectors delta(omega-omega0) with the associated
eigenvalues i*omega0. That means all eigenvalues are necessarily
imaginary, with exception of omega0=0. On the other hand, exp(t) is
an eigenvector of d/dt with eigenvalue 1, which is not part of the
spectrum of the frequency domain representation.
This means, there is no analytic continuation from other transforms,
no regularisation or transform in a weaker distributional sense.
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? ;)
This is just to illustrate that the eigenspace reasoning might not work
for infinite sums. And I don't know the sufficient condition for it to
work (and this condition wouldn't hold here anyway, probably). Or is
there any mistake in the above?
Regards,
Vadim
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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