# Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know nothing about it anymore.

We still have $F(k)$. What can we say about $f(t)$? Say for the sake of argument that the spectrum contains billions of nonzero values and thus we cannot compute it. We need a priori arguments.

As an example of the kind of insight I'm looking for, we can say whether the function $f(t)$ is even or odd just by looking at the spectrum. (If the spectrum is all real $f(t)$ is even, if all imaginary, $f(t)$ is odd.)

I am particularly interested in arguments stating whether or not there exists a $t$ such that $f(t) = x$ for some point $x$. Only given $F(k)$, can we definitively rule out or include some points $x$?

Finally, are there any results or sources in the more general, abstract literature of harmonic analysis, spectral theory, or functional analysis which could help here?

• I think I'm missing something. Why can you not perform an inverse fourier transform to recover the function? – muaddib Aug 29 '15 at 12:23
• You can certainly write the function as the inverse transform, but this will not necessarily let you talk about the function unless you 1. evaluate or 2. have the kind of insight I'm looking for. But the real answer (why I posed the question like this) is that I am looking to prove things about functions (e.g. that they do or do not take certain values) based on things I know about their spectra. – Wapiti Aug 29 '15 at 12:29
• One thing I'm thinking about is for example in the case of the Laplace transform, one has the initial and final value theorems, where (under certain conditions) one can determine the asymptotic behavior of the original function by operating on the transform: fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html And the Laplace transform is essentially a specific case of the Fourier transform. – Bitrex Aug 29 '15 at 14:50
• In addition, if the Laplace transform of a function has no roots in the right hand side of the complex plane, one can say definitively that the original function must in the limit be non-oscillating, i.e. it's not an infinite-energy signal. – Bitrex Aug 29 '15 at 14:55
• Cool, thanks. This is the kind of thing I'm looking for. – Wapiti Aug 29 '15 at 14:55