We study series that can be used to describe bounded functions on the line. The basis elements are $\exp(i \pi n x)$ and $exp (i\pi m/x)$, where $m$ and $n$ are integers with $m=0$ excluded. It turns out that this basis is stable and the coefficients are unique and may be computed in a fashion analogous to Fourier series. Due to the connection with the classical hyperbolic PDE known as the Klein-Gordon equation, we call this series a hyperbolic Fourier series.