It is known that if the nth moment of a random variable X exists, then its characteristic function is n times continuously differentiable.
My intuition here is that we can think of the distribution of X as the Fourier transform of its characteristic function (typically people talk about the characteristic function as the Fourier transform of the distribution, but my understanding is that either "direction" works with a little bookkeeping). If high moments of your distribution exists, you must have pretty fast decay in your tails. Thinking about the distribution as a Fourier transform, this is like saying that the characteristic function has very dominant low order frequencies. But of course, low order frequencies are smoother than higher order frequencies, so it makes sense then that our characteristic function would be smoother as its FT has high moments.
However, the converse of this statement is not true, which is throwing me off a bit in terms of my intuition for what is going on. For the converse not to be true, it seems as though it would need to be possible to have a very smooth function that still has nontrivial high frequencies. This is non-obvious to me. Is there an illustrative example here, or a intuitions-level explanation that could be afforded as to why the converse is not true?