The authors define a family of functions by starting with (complex) exponentials and closing under some basic algebraic operations, integration, and solution of certain systems of differential equations. They then show that for every recursively (computably) enumerable set $S$ -- in particular, even when $S$ is not computable -- there exists a function $f$ in the family whose Fourier coefficients int_-pi^pif(x),e^-inxdx are nonzero for precisely those $n$ in $S$. The paper concludes with some speculative remarks regarding hypercomputation.