Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Best approximation by subharmonic functions

Authors: J. M. Wilson and D. Zwick
Journal: Proc. Amer. Math. Soc. 114 (1992), 897-903
MSC: Primary 41A30; Secondary 31B05, 41A50
MathSciNet review: 1092929
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega \subset {\mathbb{R}^d}$ be a bounded domain. We prove existence of best subharmonic approximations in $ {L_\infty }(\Omega )$ and, for functions continuous in $ \overline \Omega $, we characterize best continuous subharmonic approximations.

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

  • [1] 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
  • [2] E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
  • [3] Frank Deutsch, Existence of best approximations, J. Approx. Theory 28 (1980), no. 2, 132–154. MR 573328,
  • [4] J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258
  • [5] Stephen D. Fisher and Joseph W. Jerome, Minimum norm extremals in function spaces, Lecture Notes in Mathematics, Vol. 479, Springer-Verlag, Berlin-New York, 1975. With applications to classical and modern analysis. MR 0442780
  • [6] 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
  • [7] 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
  • [8] 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
  • [9] L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience A Division of John Wiley & Sons, New York-London-Sydney, 1969. MR 0261018
  • [10] L. Hörmander, The analysis of linear partial differential operators I, Springer-Verlag, Berlin, Heidelberg, and New York, 1983.
  • [11] Vasant A. Ubhaya, Uniform approximation by quasi-convex and convex functions, J. Approx. Theory 55 (1988), no. 3, 326–336. MR 968939,
  • [12] D. Zwick, Best approximation by convex functions, Amer. Math. Monthly 94 (1987), no. 6, 528–534. MR 935417,
  • [13] -, The obstacle problem and best subharmonic approximation, Multivariate Approximation and Interpolation (W. Haussmann and K. Jetter, eds.), ISNM 94, Birkhäuser, 1990, pp. 313-324.

Similar Articles

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

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

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society