On Three Balls Theorem for Discrete Harmonic Functions

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• 作者：Maru Guadie (1)
  Eugenia Malinnikova (1)
  
• 关键词：Discrete Laplace operator ; Three Balls Theorem ; Logarithmic convexity ; Lagrange interpolation ; Chebyshev nodes ; 65N22 ; 65N12 ; 35J05
• 刊名：Computational Methods and Function Theory
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：14
• 期：4
• 页码：721-734
• 全文大小：197 KB
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• 作者单位：Maru Guadie (1)
  Eugenia Malinnikova (1)
  
  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491聽, Trondheim, Norway
  
• ISSN：2195-3724

文摘

We give an elementary argument to prove the Three Balls Theorem for continuous harmonic functions in \({\mathbb {R}}^n\) which can be adapted to the case of discrete harmonic functions on the lattice. The discrete analog of the Three Balls Theorem that we obtain contains an additional term that depends on the mesh size of the lattice and goes to zero when the mesh size goes to zero. We also show that any discrete harmonic function on a cube coincides with a discrete harmonic polynomial on this cube.