Reconstruction of polyharmonic functions from samples

Abstract

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.