Are there any non-constant "bump functions" in "closed form" whose Fourier Transforms are also in closed form?

Solution 1:

If your smooth function is defined on the whole real line only using $+,-,\times,\div$, and finitely many of those functions (polynomials, exponentials, trigs, and their inverses) and without a piecewise definition, and without an infinite sum or integral or whatever, then your function is better than smooth: it’s analytic. But analytic and compact support implies identically zero by the identity theorem. So the only such function is the trivial function.

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