# Literature request - finite Fourier series expansion of bounded, periodic, differentiable functions?

Question - Is someone aware of a published work proving the following: Every bounded, differentiable, periodic function of $$n$$ variables has a Fourier Series expansion with exactly $$m$$ terms ($$m$$ is finite).

I suspect this is true for n = 1, and suspect it is false for n > 1. If this is not already in a textbook somewhere, I'll attempt to (dis)prove it to myself... feels like it is "common knowledge" in a PDE textbook somewhere.

I tried searching through the links in Is there a canonical database of theorems? and there's plenty on series, but this question might be too specific.