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Transactions of the American Mathematical Society

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Best uniform approximation by solutions of elliptic differential equations


Authors: P. M. Gauthier and D. Zwick
Journal: Trans. Amer. Math. Soc. 342 (1994), 359-374
MSC: Primary 41A50; Secondary 31B99, 35B99
DOI: https://doi.org/10.1090/S0002-9947-1994-1152322-9
MathSciNet review: 1152322
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Abstract: We investigate best uniform approximations to continuous functions on compact subsets of $ {\mathbb{R}^n}$ by solutions of elliptic differential equations and, in particular, by harmonic functions. An axiomatic setting general enough to encompass problems of this kind is given, and in this context we extend necessary and sufficient conditions for best harmonic approximation on precompact Jordan domains to arbitrary compact sets and to more general classes of solutions of linear elliptic differential equations.


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  • [Ag] S. Agmon, Unicité et convexité dans les problèmes différentiels, Presses Univ. Montréal, Montréal, 1966. MR 0252808 (40:6025)
  • [BG] T. Bagby and P. M. Gauthier, Approximation by harmonic functions on closed subsets of Riemann surfaces, J. Analyse Math. 51 (1988), 259-284. MR 963157 (89j:30064)
  • [Bu] H. G. Burchard, Best uniform harmonic approximation, Approximation Theory II (G. G. Lorentz, C. K. Chui, L. L. Schumaker, eds.), Academic Press, San Diego, 1976, pp. 309-314. MR 0430631 (55:3636)
  • [Ch] E. W. Cheney, Introduction to approximation theory, McGraw-Hill, New York, 1966. MR 0222517 (36:5568)
  • [DGO] A. Dufresnoy, P. M. Gauthier, and W. H. Ow, Uniform approximation on closed sets by solutions of elliptic partial differential equations, Complex Variables Theory Appl. 6 (1986), 235-247. MR 871732 (88b:35086)
  • [GL] N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $ {A_p}$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245-268. MR 833393 (88b:35059)
  • [GGO] P. M. Gauthier, M. Goldstein, and W. H. Ow, Uniform approximation on closed sets by harmonic functions with Newtonian singularities, J. London Math. Soc. 28 (1983), 71-82. MR 703466 (84j:31009)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, Heidelberg, New York, 1977. MR 0473443 (57:13109)
  • [Ha] W. Haussmann, Approximation by harmonic functions, Topics in Multivariate Approximation (C. K. Chui, L. L. Schumaker, F. I. Utreras, eds.), Academic Press, San Diego, 1987, pp. 111-124. MR 924826 (89b:31007)
  • [HZ] W. Haussmann and K. Zeller, H-sets and best uniform approximation by solutions of elliptic differential equations, Results in Math. 4 (1988), 84-92. MR 956007 (89k:41032)
  • [HKL] 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, ISNM 65 (P. L. Butzer, R. L. Stens, and B. Sz.-Nagy, eds.), Academic Press, San Diego, 1984, pp. 317-327. MR 820533 (87i:41021)
  • [He] L. L. Helms, Introduction to potential theory, Wiley-Interscience, New York, 1969. MR 0261018 (41:5638)
  • [HY] J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, MA, 1961. MR 0125557 (23:A2857)
  • [Hö] L. Hörmander, The analysis of linear partial differential operators. I, Springer-Verlag, Berlin, Heidelberg, New York, 1983. MR 717035 (85g:35002a)
  • [In] E. L. Ince, Ordinary differential equations, Longmans, Green, London, 1927.
  • [KS] S. Karlin and W. Studden, Tchebycheff systems: with applications in analysis and statistics, Wiley-Interscience, New York, 1966. MR 0204922 (34:4757)
  • [La] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, Berlin, Heidelberg, New York, 1972. MR 0350027 (50:2520)
  • [Na1] R. Narasimhan, Complex analysis in one variable, Birkhäuser, Boston and Basel, 1985. MR 781130 (87h:30001)
  • [Na2] -, Analysis on real and complex manifolds, North-Holland, Amsterdam, 1968. MR 0251745 (40:4972)
  • [PW] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1967. MR 0219861 (36:2935)
  • [SZ] S. Saks and A. Zygmund, Analytic functions, Monogr. Mat., Warsaw, 1952. MR 0055432 (14:1073a)
  • [Si] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, Berlin, Heidelberg, New York, 1970. MR 0270044 (42:4937)
  • [Wi] J. M. Wilson, A counterexample in the theory of best approximation, J. Approx. Theory (to appear). MR 1081038 (91i:41023)
  • [Zw] D. Zwick, Uniform approximation by harmonic and subharmonic functions, Approximation Theory VI (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, Boston, 1989, pp. 689-692. MR 1091082 (92c:41030)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1152322-9
Article copyright: © Copyright 1994 American Mathematical Society

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