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Computational Methods and Function Theory

, Volume 14, Issue 4, pp 721–734 | Cite as

On Three Balls Theorem for Discrete Harmonic Functions

  • Maru Guadie
  • Eugenia Malinnikova
Article

Abstract

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.

Keywords

Discrete Laplace operator Three Balls Theorem Logarithmic convexity Lagrange interpolation Chebyshev nodes 

Mathematics Subject Classification (2000)

65N22 65N12 35J05 

Notes

Acknowledgments

The authors would like to thank the Center of Advanced Study at the Norwegian Academy of Science and Letters in Oslo where the work was completed. This work is a part of the first author’s Ph.D. thesis. It is our pleasure to thank the external members of the evaluation committee of the thesis Alexander Rashkovskii and Eero Saksman for their useful questions and remarks.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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