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I wonder if a situation is possible where, we measure some signal/property concerning a black hole. Supposing the measurement we make with some telescope, gets us the Fourier transform coefficients of the signal, and then by using the Fourier coefficients...we make some calculations which suggest a possibility of a singularity in the signal/parameter. I know its not possible to confirm a singularity by finite computations experimentally but I guess its innteresting to see the data suggesting something like a singularity. I hope to use this result on the measured Fourier coefficients.

All I need is a data measurement of this type, if possible and appreciate some suggestions of possibility.

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This is actually more subtle than you would think it would be. First, remember that a fourier transform is defined, for some time-dependent signal $F(t)$, as${}^{1}$:

$$F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} dt\, e^{i\omega\,t} F(t)$$

Well, this is great in special relativity, but in general relativity, what time do we actually use? The natural choices are the time of the distant observer that is taking the measurment, or the time as measured by some local, freely falling observer. It turns out that these choices are inequalvent, and that, in fact, this is the heart of one of the mechanisms by which you can find Hawking radiation${}^{2}$.

All of this is a long way of saying that the means by which GR rotates time into space will prevent us from "integrating" over the portion of the field that enters into the black hole horizon. There is no way to use Fourier transforms to probe into the whole. No causal influence can come out of the hole, at least classically.

${}^{1}$If we're being properly covariant, I guess I should write this as $F(x^{a})$, and have the transform be $F(k^{a}) $, and have the exponent be $e^{ik^{a}x_{a}}$, but this basic problem still persists.

${}^{2}$Remember that the definition of the vacuum state, and thus the whole Fock space of quantum field theory, is based upon taking Fourier transforms of the fields, finding a basis of the space of free fields, and then defining normal ordering and counting operators. What is a "particle" and what is an "anti-particle" depends on these processes.

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