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Mean values and differential equations

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Abstract

We show that the well-known equivalence between the mean-value theorem and harmonicity extends to arbitrary measures of compact support: a continuous function satisfies the generalized mean-value condition (1) with respect to a given measure if and only if it is annihilated by a certain system of homogeneous linear partial differential operators with constant coefficients determined by the measure. Extensions of this result are obtained, primarily in the direction of replacing systems of differential equations by a single equation.

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References

  1. J. Aczél, H. Haruki, M. A. McKiernan, and G. N. Sakovič,General and regular solutions of functional equations characterizing harmonic polynomials, Aequationes Math.1 (1968), 37–53.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. G. Arsove,The Looman-Menchoff theorem and some subharmonic function analogues, Proc. Amer. Math. Soc.6 (1955), 94–105.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. A. Baker,An analogue of the wave equation and certain related equations, Canad. Math. Bull.12 (1969), 837–846.

    MATH  MathSciNet  Google Scholar 

  4. E. F. Beckenbach and M. Reade,Mean values and harmonic polynomials, Trans. Amer. Math. Soc.53 (1943), 230–238.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. F. Beckenbach and M. Reade,Regular solids and harmonic polynomials, Duke Math. J.12 (1945), 629–644.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. A. Berenstein and M. A. Dostal,Analytically Uniform Spaces and their Applications to Convolution Equations, Springer-Verlag, 1972.

  7. J. Bourget,Mémoire sur le mouvement vibratoire des membranes circulaires, Ann. Sci. Ecole Norm. Sup.3 (1866), 55–95.

    MathSciNet  Google Scholar 

  8. J. H. Bramble and L. E. Payne,Mean value theorems for polyharmonic functions, Amer. Math. Monthly73 (1966), Part II, 124–127.

    Article  MathSciNet  Google Scholar 

  9. W. Brödel,Funktionen mit Gaussischer Mittelwerteigenschaften für konvexe Kurven und Bereiche, Deutsche Mathematik4 (1939), 3–15.

    MATH  Google Scholar 

  10. Min-Teh Cheng,On a theorem of Nicolesco and generalized Laplace operators, Proc. Amer. Math. Soc.2 (1951), 77–86.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Choquet and J. Deny,Sur quelques propriétés de moyenne charactérstiques des fonctions harmoniques et polyharmoniques, Bull. Soc. Math. France72 (1944), 118–140.

    MATH  MathSciNet  Google Scholar 

  12. R. Courant and D. Hilbert,Methods of Mathematical Physics, Vol. 2, Interscience, 1962.

  13. L. Ehrenpreis,Fourier Analysis in Several Complex Variables, Interscience, 1970.

  14. I. Fenyö,Remark on a paper of J. A. Baker, Aequationes Math.8 (1972), 103–108.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Flatto,Functions with a mean value property, J. Math. Mech.10 (1961), 11–18.

    MATH  MathSciNet  Google Scholar 

  16. L. Flatto,Functions with a mean value property II, Amer. J. Math.85 (1963), 248–270.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Flatto,Partial differential equations and difference equations, Proc. Amer. Math. Soc.16 (1965). 858–863.

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Flatto,On polynomials characterized by a certain mean value property, Proc. Amer. Math. Soc.17 (1966), 598–601.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Friedman and W. Littman,Functions satisfying the mean value property, Trans. Amer. Math. Soc.102 (1962), 167–180.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Garsia,A note on the mean value property, Trans. Amer. Math. Soc.102 (1962), 181–186.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. M. Garsia and E. Rodemich,On functions satisfying the mean value property with respect to a product measure, Proc. Amer. Math. Soc.17 (1966), 592–594.

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Hörmander,Linear Partial Differential Operators, Springer-Verlag, 1964.

  23. J. E. Littlewood,A Mathematican's Miscellany, Methuen, 1953.

  24. M. A. McKiernan,Boundedness on a set of positive measure and the mean value property characterizes polynomials on a space V n, Aequations Math.4 (1970), 31–36.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Nicolesco,Les Fonctions Polyharmoniques, Hermann, 1936.

  26. V. P. Palamodov,Linear Differential Operators with Constant Coefficients, Springer-Verlag, 1970.

  27. P. Pizzetti,Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rend. Lincei, ser. 5, 18 (1909), 182–185.

    Google Scholar 

  28. P. Pizzetti,Sul significato geometrico del secundo parametro differenziale di una funzione sopra una superficie qualunque, Rend. Lincei, ser. 5,18, (1909), 309–316.

    Google Scholar 

  29. H. Poritsky,Generalizations of the Gauss law of the spherical mean, Trans. Amer. Math. Soc.43 (1938), 199–225.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Schwartz,Théorie des Distributions, Vols. 1 and 2, Hermann, 1957.

  31. V. L. Shapiro,Sets of uniqueness for the vibrating string problem Trans. Amer. Math. Soc.141 (1969), 127–146.

    Article  MATH  MathSciNet  Google Scholar 

  32. C. L. Siegel,Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys-math. Kl. 1929, no. 1.

  33. H. Šwiatak,Criteria for the regularity of continuous and locally integrable solutions of a class of linear functional equations, Aequations Math.6 (1971), 170–187.

    Article  MATH  Google Scholar 

  34. F. Trèves,Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966.

  35. B. L. van der Waerden,Modern Algebra, Vols. 1, and 2, Frederich Ungar, 1953, and 1950.

  36. V. Volterra,Alcune osservazioni sopra proprietà atte ad individuare una funzione, Rend. Lincei, ser. 5,18 (1909), 263–266.

    Google Scholar 

  37. J. L. Walsh,A mean value theorem for polynomials and harmonic polynomials, Bull. Amer. Math. Soc.42 (1936), 923–930.

    Article  MATH  MathSciNet  Google Scholar 

  38. G. N. Watson,A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962.

  39. H. Weber, Ueber die Integration der partiellen Differentialgleichung: {ie352-1}, Math. Ann.1 (1869), 1–36.

    Article  MathSciNet  Google Scholar 

  40. L. Zalcman,Analyticity and the Pompeiu problem, Arch. Rat. Mech. Anal.47 (1972), 237–254.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Maryland, College Park, Maryland, U.S.A.

    Lawrence Zalcman

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  1. Lawrence Zalcman
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Additional information

Research and preparation supported in part by NSF GP 28970.

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Zalcman, L. Mean values and differential equations. Israel J. Math. 14, 339–352 (1973). https://doi.org/10.1007/BF02764713

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  • DOI: https://doi.org/10.1007/BF02764713

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