  Fourier extensions have been shown to be an effective means for the
approximation of smooth, nonperiodic functions on bounded intervals given their
values on an equispaced, or in general, scattered grid. Related to this method
are two parameters. These are the extension parameter $T$ (the ratio of the
size of the extended domain to the physical domain) and the oversampling ratio
$\eta$ (the number of sampling nodes per Fourier mode). The purpose of this
paper is to investigate how the choice of these parameters affects the accuracy
and stability of the approximation. Our main contribution is to document the
following interesting phenomenon: namely, if the desired condition number of
the algorithm is fixed in advance, then the particular choice of such
parameters makes little difference to the algorithm's accuracy. As a result,
one is free to choose $T$ without concern that it is suboptimal. In particular,
one may use the value $T=2$ - which corresponds to the case where the extended
domain is precisely twice the size of the physical domain - for which there is
known to be a fast algorithm for computing the approximation. In addition, we
also determine the resolution power (points-per-wavelength) of the
approximation to be equal to $T \eta$, and address the trade-off between
resolution power and stability.
