Reconstruction of polyharmonic functions from samples

We study planar complex-valued functions that satisfy a certain Wirtinger differential equation of order k. Our considerations include entire functions (k=1), harmonic functions (k=2), biharmonic functions (k=4), and polyharmonic functions (k even) in general. Under the assumption of restricted exponential growth and square integrability along the real axis, we establish a sampling theorem that extends the classical sampling theorem of Whittaker–Kotel’nikov–Shannon and reduces to the latter when k=1. Intermediate steps, which may be of independent interest, are representation theorems, uniqueness theorems, and the construction of fundamental functions for interpolation. We also consider supplements, variants, generalizations, and an algorithm.