Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Myron Goldstein, Werner Haussmann and Lothar Rogge
Journal: Trans. Amer. Math. Soc. 305 (1988), 505-515
MSC: Primary 31B05; Secondary 41A30, 41A50
MathSciNet review: 924767
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

    M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical functions, Dover, New York, 1972.
  • S. N. Bernšteĭn, Sobranie sočinenii. Tom I. Konstruktivnaya teoriya funkciĭ [1905–1930], Izdat. Akad. Nauk SSSR, Moscow, 1952 (Russian). MR 0048360
  • M. Brelot, Sur l’approximation et la convergence dans la théorie des fonctions harmoniques ou holomorphes, Bull. Soc. Math. France 73 (1945), 55–70 (French). MR 13824
  • H. G. Burchard, Best uniform harmonic approximation, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 309–314. MR 0430631
  • E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
  • Jacques Deny, Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier (Grenoble) 1 (1949), 103–113 (1950) (French). MR 37414
  • Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
  • 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), no. 1, 71–82. MR 703466, DOI
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR 0473443
  • M. Goldstein, W. Haussmann, and K. Jetter, Best harmonic $L^1$ approximation to subharmonic functions, J. London Math. Soc. (2) 30 (1984), no. 2, 257–264. MR 771421, DOI
  • W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathematical Society Monographs, No. 9. MR 0460672
  • Walter K. Hayman, Donald Kershaw, and Terry J. Lyons, The best harmonic approximant to a continuous function, Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 65, Birkhäuser, Basel, 1984, pp. 317–327. MR 820533
  • Lars Inge Hedberg, Approximation in the mean by solutions of elliptic equations, Duke Math. J. 40 (1973), 9–16. MR 312071
  • E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea, New York, 1965.
  • Ü. Kuran, On the mean-value property of harmonic functions, Bull. London Math. Soc. 4 (1972), 311–312. MR 320348, DOI
  • G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
  • John C. Polking, Approximation in $L^{p}$ by solutions of elliptic partial differential equations, Amer. J. Math. 94 (1972), 1231–1244. MR 324215, DOI
  • John R. Rice, The approximation of functions. Vol. I: Linear theory, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964. MR 0166520
  • Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Translated from the Romanian by Radu Georgescu. MR 0270044
  • Richard L. Wheeden and Antoni Zygmund, Measure and integral, Marcel Dekker, Inc., New York-Basel, 1977. An introduction to real analysis; Pure and Applied Mathematics, Vol. 43. MR 0492146

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31B05, 41A30, 41A50

Retrieve articles in all journals with MSC: 31B05, 41A30, 41A50

Additional Information

Keywords: Harmonic and subharmonic functions, inverse mean value property, harmonic <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^1}$">-approximation
Article copyright: © Copyright 1988 American Mathematical Society