If we have real functions such that $f \in L^1$ and $g \in L^2$, do we still have always the following equality, providing left hand-side is well defined (Plancherel theorem ?) :
$$\int_{-\infty}^{\infty} f(x) \overline{g(x)} dx = \int_{-\infty}^{\infty} \hat{f}(x) \overline{\hat{g}(x)} dx$$
(theorem is often given for $f$ and $g$ in $L^1 \cap L^2$, but what is the widest form of this relation without involving distributions ?) If above equality is not always true, any counter-example ?