Can a non-periodic function have a Fourier series? Consider two periodic functions. Assume their sum is not periodic.
The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents their sum. But their sum is not periodic, yet you have described it using a Fourier series.
I thought that non-periodic functions can't be represented by a Fourier series. Why isn't this a contradiction?

To answer the original question: it's a matter of convention or terminology. The most usual usage (as visible in the other answers and comments) is that "Fourier series" refers to that of a periodic function, or an extension-by-periodicity of a function on an interval to a periodic function on the line. In particular, a sum of sines and cosines, or exponentials, with some incommensurable periods, would often not be considered a Fourier series.
Yet, certainly such a thing is a sum of cosines and sines, or exponentials, so it's related in some ways. Such things were investigated at length in the early 20th century by Harald Bohr and others, in part to look at the behavior of Dirichlet series.
So, on one hand, in some contexts "Fourier series" means "of periodic function". In others, it might not.
