Reconstruction of polyharmonic functions from samples

  • Authors:

  • Gerhard Schmeisser

  • Affiliations:

  • Mathematical Institute, University of Erlangen-Nuremberg, 91054 Erlangen, Germany

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2007

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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.