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On the mean value property of harmonic functions and best harmonic $ L\sp 1$-approximation


Authors: Myron Goldstein, Werner Haussmann and Lothar Rogge
Journal: Trans. Amer. Math. Soc. 305 (1988), 505-515
MSC: Primary 31B05; Secondary 41A30, 41A50
DOI: https://doi.org/10.1090/S0002-9947-1988-0924767-8
MathSciNet review: 924767
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Abstract: The present paper deals with the inverse mean value property of harmonic functions and with the existence, uniqueness, and characterization of a best harmonic $ {L^1}$-approximant to strictly subharmonic functions. The main theorem concerning the inverse mean value property of harmonic functions is based on a generalization of a theorem due to Ü. Kuran as well as on an approximation theorem proved by J. C. Polking and also by L. I. Hedberg. The inverse mean value property will be applied in order to prove necessary and sufficient conditions for the existence of a best harmonic $ {L^1}$-approximant to a subharmonic function $ s$ satisfying $ \Delta s > 0$ a.e. in the unit ball.


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  • [1] M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical functions, Dover, New York, 1972.
  • [2] S. N. Bernstein, Collected works, Vol. 1, Izdat. Akad. Nauk SSSR, Moscow, 1952. MR 0048360 (14:2c)
  • [3] M. Brelot, Sur l'approximation et la convergence dans la théorie des fonctions harmoniques ou holomorphes, Bull. Soc. Math. France 73 (1945), 55-70. MR 0013824 (7:205a)
  • [4] H. G. Burchard, Best uniform harmonic approximation, Approximation Theory II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, eds.), Academic Press, New York and London, 1976, pp. 309-314. MR 0430631 (55:3636)
  • [5] E. W. Cheney, Introduction to approximation theory, McGraw-Hill, New York, 1966. MR 0222517 (36:5568)
  • [6] J. Deny, Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier (Grenoble) 1 (1949), 103-113. MR 0037414 (12:258c)
  • [7] T. W. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. MR 0410387 (53:14137)
  • [8] P. M. Gauthier, M. Goldstein and W. H. Ow, Uniform approximation on closed sets by harmonic functions with Newtonian singularities, J. London Math. Soc. (2) 28 (1983), 71-82. MR 703466 (84j:31009)
  • [9] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, S[ill]nger, Berlin, Heidelberg and New York, 1977. MR 0473443 (57:13109)
  • [10] M. Goldstein, W. Haussmann and K. Jetter, Best harmonic $ {L^1}$ approximation to subharmonic functions, J. London Math. Soc. (2) 30 (1984), 257-264. MR 771421 (86b:31002)
  • [11] W. K. Hayman and P. B. Kennedy, Subharmonic functions, Academic Press, London and New York, 1976. MR 0460672 (57:665)
  • [12] W. K. Hayman, D. Kershaw and T. J. Lyons, The best harmonic approximant to a continuous function, Anniversary Volume on Approximation Theory and Functional Analysis (P. L. Butzer, R. L. Stens, B. Sz.-Nagy, eds.), Internat. Ser. Numer. Math. 65, Birkhäuser, Basel, Boston and Stuttgart, 1984, pp. 317-327. MR 820533 (87i:41021)
  • [13] L. I. Hedberg, Approximation in the mean by solutions of elliptic equations, Duke Math. J. 40 (1973), 9-16. MR 0312071 (47:633)
  • [14] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea, New York, 1965.
  • [15] Ü. Kuran, On the mean value property of harmonic functions, Bull. London Math. Soc. 4 (1972), 311-312. MR 0320348 (47:8887)
  • [16] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 0213785 (35:4642)
  • [17] J. C. Polking, Approximation in $ {L^p}$ by solutions of elliptic partial differential equations, Amer. J. Math. 94 (1972), 1231-1244. MR 0324215 (48:2567)
  • [18] J. R. Rice, The approximation of functions, Vol. 1, Addison-Wesley, Reading, Mass., 1964. MR 0166520 (29:3795)
  • [19] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer, Berlin, Heidelberg and New York, 1970. MR 0270044 (42:4937)
  • [20] R. L. Wheeden and A. Zygmund, Measure and integral: An introduction to real analysis, Dekker, New York and Basel, 1977. MR 0492146 (58:11295)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0924767-8
Keywords: Harmonic and subharmonic functions, inverse mean value property, harmonic $ {L^1}$-approximation
Article copyright: © Copyright 1988 American Mathematical Society

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