Epicycles as precursors of Fourier series

Regarding "precursor", depends on exactly what you mean, as Eric Tressler pointed out.

But in fact epicycles are an example of "nonharmonic Fourier series"; that is, sums like $ae^{i\alpha t}+be^{i\beta t}$ where $\alpha/\beta$ may be irrational.

And what's interesting about that is that there's a theorem saying Ptolemy was right! In a sense, sort of. There are various different flavors of "almost periodic function" out there. I think the most common and well known is what that page calls "Uniform or Bohr or Bochner" almost periodic functions. There's an intrinsic definition in terms of "almost periods", and it's a theorem that any almost periodic function in this sense is a uniform limit of (non-harmonic) trigonometric polynomials.

So all one has to do is demonstrate that planetary orbits are almost periodic and it follows that they can be described arbitrarily well using epicycles. Cool.

You might want to take a look at the paper Epicycles are Almost as Good as Trigometric Series by Acosta, Smith, Kosheleva & Kreinovich

They report that epicycles were originally proposed by Appollonius of Perga, late 3rd to early 2nd century BCE, developed by several others, and finalised by Ptolomy in the 1st century AD.

They remark that traditional textbooks term it as 'bad science' that was overthrown by the 'genius' of Copernicus.

But that from the mathematical sense a system of epicycles make perfect sense as a trigonometric series.

They don't comment on any direct connection between epicycles and Fourier series. However, it is known that Fourier was interested in physics - and so likely, astronomy.