A characterization of $L^1$ functions with $L^1$ Fourier transforms

Question

If $f \in L^1$ then its Fourier transform is easy to define: $\widehat{f}(\xi) = \int f(x)e^{-2 \pi i \xi x}dx$. If $\widehat{f} \in L^1$ then we recover $f(x) = \int \widehat{f}(\xi)e^{2 \pi i \xi x} d \xi$, a nice formula; if not, there's the not-so-nice formula $\|f \ast K_{\delta} - f\|_1 \to 0$ as $\delta \to 0$ (where $K_{\delta}$ is the Gauss kernel), and $f \ast K_{\delta}$ can be written in terms of $\widehat{f}$. My question is: is there a nice characterization of when I can apply the first formula, that is, which $L^1$ functions have $L^1$ Fourier transforms? If not, what sorts of classes of functions satisfy this? The Schwartz class would be such an example.

Answer 1

As commenters said, there is no explicit characterization of functions with an integrable Fourier transforms. A natural and useful class of functions that have integrable Fourier transforms is the Sobolev space $H^s(\mathbb R^n)$ with $s>n/2$: $$f\in H^s \iff f\in L^2 \text{ and } |\xi|^s \hat f(\xi) \in L^2 $$ Indeed, for such functions $$\|\hat f\|_{L^1} = \int_{\mathbb R^n} \Big((1+|\xi|^s) \hat f(\xi) \Big) \cdot \frac{1}{1+|\xi|^s}\,d\xi <\infty $$ because both factors in the last integral are in $L^2$.

In classical terms: if $f$ is square integrable and so are its $k$th order derivatives for some $k>n/2$, then the above applies.