Suppose some function is approximated by a Taylor Series. Then the Taylor Series is approximated by a Fourier Series, which would entail approximating each term in the Taylor Series by its own Fourier Series. Then each term in each of the Fourier Series is approximated by a Taylor Series, and each of those terms by a Fourier Series, and the pattern is repeated indefinitely.
Obviously this would be a huge algebraic mess, but what would the result look like geometrically? Would it resemble the original function? Would it converge to a single, special geometry independent of the original function? Would it matter if a Fourier Series was applied as the first step instead of a Taylor Series?