I am looking for a proof of a result which Bochner and Chandrasekharan mention as a side remark in their book "Fourier Transforms", pg 13, on uniqueness of Fourier transforms. After proving uniqueness of FT under standard hypotheses with essentially Polya's proof, it is stated that
"In one variable, there is a deeper theorem in which the (absolute) integrability of $f(x)$ in $(-\infty, +\infty)$ is entirely dispensed with and only the existence of the Cauchy limit $$ \phi(\alpha) = \lim_{A \to +\infty} \int_{-A}^A e^{i\alpha x} f(x) dx $$ for every $\alpha$ is presupposed. This type of theorem has, apparently, never been extended to more than one variable and any non-trivial result in this direction would be very welcome indeed".
I have been unable to find a proof or references for this result, which seems very subtle. Any help would be most welcome.