Fourier transform of locally integrable function

Question

I have a question and I don't know if it is true. Is any locally integrable function a sum of an $L_{1}$ function and another nice function (perhaps, an $L_{2}$ function). This is related to the Fourier transform of a locally integrable function.

When can we take the Fourier transform of an locally integrable function? (What condition do we need for an $L_{1,loc}$ such that the Fourier transform exists?)

If my question was true, then it yields that we can always take it, which I am skeptical about. Thank you.

Answer 1

Negative, $f(x)=e^x$ cannot be represented as a sum of $L^1$ function with anything substantially better than $f$ itself.

The relevant topic is tempered distributions. In a nutshell, if a locally integrable function has at most polynomial growth at infinity, then its Fourier transform can be defined -- but not as a function in general, as a distribution.