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Mean-Value Theorems for Harmonic Functions on the Cube in R<sup><i>n</i></sup>
Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria
- 1 Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria
Advances in Pure Mathematics·Volume 05 (2015)·Pages 683–688·Published 7 September 2015·DOI10.4236/apm.2015.511062
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Abstract
Let be a hypercube in R n . We prove theorems concerning mean-values of harmonic and polyharmonic functions on I n ( r ), which can be considered as natural analogues of the famous Gauss surface and volume mean-value formulas for harmonic functions on the ball in and their extensions for polyharmonic functions. We also discuss an application of these formulas—the problem of best canonical one-sided L1-approximation by harmonic functions on I n ( r ).
KeywordsHarmonic FunctionsPolyharmonic FunctionsHypercubeQuadrature DomainBest One-Sided Approximation
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