  We consider the problem of approximating a function using Herglotz wave
functions, which are a superposition of plane waves. When the discrepancy is
measured in a ball, we show that the problem can essentially be solved by
considering the function we wish to approximate as a source distribution and
time reversing the resulting field. Unfortunately this gives generally poor
approximations. Intuitively, this is because Herglotz wave functions are
determined by a two-dimensional field and the function to approximate is
three-dimensional. If the discrepancy is measured on a plane, we show that the
best approximation corresponds to a low-pass filter, where only the spatial
frequencies with length less than the wavenumber are kept. The corresponding
Herglotz wave density can be found explicitly. Our results have application to
designing standing acoustic waves for self-assembly of micro-particles in a
fluid.
