Reconstruction of polyharmonic functions from samples

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• 作者：Gerhard Schmeisser
• 关键词：Polyharmonic functions ; Wirtinger calculus ; Representation theorems ; Uniqueness theorems ; Fundamental functions ; Sampling ; Reconstruction formulae
• 刊名：Journal of Approximation Theory
• 出版年：2007
• 出版时间：September 2007
• 年：2007
• 卷：148
• 期：1
• 页码：49-70
• 全文大小：260 K

文摘

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 (cdca3fdb"" title=""Click to view the MathML source"">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.