the smoothness-tails duality
A recurring theme of Fourier analysis is a trade-off: how smooth a function is at the spatial side controls how fast its transform decays at the frequency side, and vice versa. For probability this becomes a dictionary translating tail behaviour of a distribution into smoothness of its characteristic function, and smoothness of the density into decay of the cf. It is the qualitative intuition behind moment-derivative relations, the local CLT, and Edgeworth corrections.
Two paired statements capture it. First, decay of phi controls smoothness of the law: if the cf phi is integrable, i.e. integral |phi(t)| dt < infinity, then by Fourier inversion the law has a bounded continuous density f(x) = (1/2pi) integral e^(-itx) phi(t) dt; faster decay of phi (say |phi(t)| <= C/(1+|t|)^(k+1)) gives a density with k continuous derivatives. Second, tails control smoothness of phi at 0: heavier tails of X make phi rougher at the origin. Concretely, E[|X|^k] < infinity forces phi to be k-times continuously differentiable, with phi^(j)(0) = i^j E[X^j]; conversely a finite even derivative phi^(2m)(0) forces E[X^(2m)] < infinity. So a heavy-tailed law (no high moments) has a cf with limited smoothness at 0 — the Cauchy cf e^(-|t|) has a corner at 0, reflecting the Cauchy's lack of even a mean.
This duality is the working intuition for refined limit theory: the existence of a smooth limiting density (local CLT) needs cf decay; Edgeworth corrections come from expanding the cf near 0 using higher moments; and heavy tails (slow phi-smoothness at 0) are exactly what break the classical CLT and push you toward stable laws. Honest caveat: the dictionary is qualitative, not a clean if-and-only-if at every order. Odd-order moments and one-sided tails behave subtly, and a cf can be non-differentiable at 0 yet correspond to a law with a finite mean defined only as a principal value; treat the moment-derivative correspondence carefully at boundary cases.
The standard normal has cf e^(-t^2/2), which decays super-fast and is infinitely smooth — matching its smooth, all-moments-finite density. The standard Cauchy has cf e^(-|t|), which has a corner at 0 (not differentiable there) — matching its heavy tails: the Cauchy has no finite mean, so phi cannot have a finite first derivative at 0.
Smooth density <-> fast cf decay; finite moments <-> cf differentiable at 0. Heavy tails leave a corner.
The duality is qualitative: faster decay of phi gives a smoother density, but an integrable cf only guarantees a continuous density, not necessarily a smooth one without extra decay.