  We present a fast and spectrally accurate numerical scheme for the evaluation
of the gyroaveraged electrostatic potential in non-periodic gyrokinetic-Poisson
simulations. Our method relies on a reformulation of the gyrokinetic-Poisson
system in which the gyroaverage in Poisson's equation is computed for the
compactly supported charge density instead of the non-periodic, non-compactly
supported potential itself. We calculate this gyroaverage with a combination of
two Fourier transforms and a Hankel transform, which has the near optimal run
time complexity $O(N_{\rho}(P+\hat{P})\log(P+\hat{P}))$, where $P$ is the
number of spatial grid points, $\hat{P}$ the number of grid points in Fourier
space, and $N_{\rho}$ the number of grid points in velocity space. We present
numerical examples illustrating the performance of our code and demonstrating
geometric convergence of the error.
