In this answer @icurays1 states:
In this same vein, Fourier analysis leads to an extremely powerful theory of smoothness, because of the correspondence between differentiability and decay of the Fourier coefficients.
Similarly, in this section of a presentation dealing with the decomposition of functions using spherical harmonics, the presenter states, "[...] spectral smoothness corresponds to spatial decay[...]"
My question is two-fold. First, does this correspondence go both ways (i.e. does spatial decay imply spectral smoothness)? Second, if the converse holds, what is the minimal condition on a function on $\mathbb{R}^n$ for the Fourier transform to exist and be continuous everywhere in Fourier space (differentiability is not needed for my purposes)? For example, is the minimal condition that the function has a finite $L^2$ norm? A finite second moment?
Apart from the Riemann-Lebesgue lemma, which asserts that $f\in L^1(\mathbb R^n$ implies that $\widehat{f}$ is continuous (and goes to $0$ at infinity), there are also $L^2$-style conditions coming from Sobolev theory.
One basic "Sobolev imbedding theorem" is that for $f$ on $\mathbb R^n$ with $|\widehat{f}|^2\cdot (1+|x|^2)^{\frac{n}{2}+\varepsilon}$ in $L^2(\mathbb R^n)$, $f$ is continuous. Reversing the roles gives an $L^2$-style condition for continuity of $\widehat{f}$.
Such basic Sobolev stuff occurs in many places: Folland's TATA notes on PDE, also Folland's fuller PDE book, Evans' PDE book, Taylor's PDE book(s) I, ... and probably Google-able. My own course notes in both Real Analysis and Functional Analysis do several of these basic things, as I'm sure many others' course notes do, as well:
http://www.math.umn.edu/~garrett/m/real/ and http://www.math.umn.edu/~garrett/m/fun/