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A generalized mean value property for polyharmonic functions

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Abstract

A well known property of a harmonic function in a ball is that its value at the centre equals the mean of its values on the boundary. Less well known is the more general property that its value at any point x equals the mean over all chords through x of its values at the ends of the chord, linearly interpolated at x. In this paper we show that a similar property holds for polyharmonic functions of any order when linear interpolation is replaced by two-point Hermite interpolation of odd degree.

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References

  1. Almansi, E.: Sulle integrazione dell’equazione differenciale Δ2n=0. Annali di Matematica 2, 1–59 (1899)

    Article  Google Scholar 

  2. Aronszajn, N., Creese, T., Lipkin, L.: Polyharmonic functions. Clarendon Press, Oxford (1983)

    MATH  Google Scholar 

  3. Belyaev, A.: On transfinite barycentric coordinates. In: Symposium on Geometry Processing 2006, Eurographics Association, pp 89–99 (2006)

  4. Conte, S.D., de Boor, C.: Elementary numerical analysis. McGraw-Hill (1980)

  5. Duffin, R.J., Nehari, Z.: Note on polyharmonic functions. Proc. Amer. Math. Soc. 12, 110–115 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  6. Edenhoffer, J.: Integraldarstellungen einer m-polyharmonischen funktion, deren funktionswerte und erste m−1 normalableitungen auf einer hypersphäre gegeben sind. Math. Nachr. 68, 105–113 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Floater, M.S.: Generalized barycentric coordinates and applications. Acta Numerica 24, 161–214 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  9. Gordon, W.J., Wixom, J.A.: Pseudo-harmonic interpolation on convex domains. SIAM J. Numer. Anal. 11, 909–933 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hayman, W.K., Korenblum, B.: Representation and uniqueness theorems for polyharmonic functions. J. d’Analyse Math. 60, 113–133 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Isaacson, E., Keller, H.B.: Analysis of numerical methods (1994)

  12. Li, X.-Y., Hu, S.-M.: Poisson coordinates. IEEE Trans. Vis. Comput. Graph. 19, 344–352 (2013)

    Article  Google Scholar 

  13. Li, X.-Y., Ju, T., Hu, S.-M.: Cubic mean value coordinates. ACM Trans. Graph. 32, 126:1–126:10 (2013)

    MATH  Google Scholar 

  14. Manson, J., Li, K., Schaefer, S.: Positive Gordon-Wixom coordinates. Computer Aided Design 43, 1422–1426 (2011)

    Article  Google Scholar 

  15. Nicolesco, M.: Nouvelles contributions dans la théorie des fonctions polyharmoniques. Bull. Math. Soc. Sei. Math. Phys. R.P. Roumaine 37, 79–95 (1935)

    MATH  Google Scholar 

  16. Nicolesco, M.: Les fonctions polyharmoniques. Actualités scientifiques et industrielles 331 (1936)

  17. Polyanin, A.D.: Handbook of linear partial differential equations for engineers and scientists. Chapman and Hall (2002)

  18. Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-spline techniques. Springer (2002)

  19. Weber, O., Poranne, R., Gotsman, C.: Biharmonic coordinates. Comput. Graph. Forum 31, 2409–2422 (2012)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851, Oslo, Norway

    Michael S. Floater

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  1. Michael S. Floater
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Correspondence to Michael S. Floater.

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Floater, M.S. A generalized mean value property for polyharmonic functions. Numer Algor 73, 157–165 (2016). https://doi.org/10.1007/s11075-015-0090-7

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  • DOI: https://doi.org/10.1007/s11075-015-0090-7

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