Fourier interpolation in dimensions 3 and 4 and real-variable Kloosterman sums

We give a construction of radial Fourier interpolation formulas in dimensions 3 and 4 using Maass–Poincaré type series. As a corollary we obtain explicit formulas for the basis functions of these interpolation formulas in terms of what we call real-variable Kloosterman sums, which were previously introduced by Stoller. We also improve the bounds on the corresponding basis functions $a_{n,d}(x)$, $d=3,4$, for fixed $x$, in terms of the index $n$.