  Analysis of radio signals from cosmic-ray induced air showers has been shown
to be a reliable method to extract shower parameters such as primary energy and
depth of shower maximum. The required detailed air shower simulations take 1 to
3 days of CPU time per shower for a few hundred antennas. With nearly $60,000$
antennas envisioned to be used for air shower studies at the Square Kilometre
Array (SKA), simulating all of these would come at unreasonable costs. We
present an interpolation algorithm to reconstruct the full pulse time series at
any position in the radio footprint, from a set of antennas simulated on a
polar grid. Relying on Fourier series representations and cubic splines, it
significantly improves on existing linear methods. We show that simulating
about 200 antennas is sufficient for high-precision analysis in the SKA era,
including e.g. interferometry which relies on accurate pulse shapes and
timings. We therefore propose the interpolation algorithm and its
implementation as a useful extension of radio simulation codes, to limit
computational effort while retaining accuracy.
