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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
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] Shmuel Agmon, Unicité et convexité dans les problèmes différentiels, Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965), Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR 0252808
  • [BG] Thomas Bagby and P. M. Gauthier, Approximation by harmonic functions on closed subsets of Riemann surfaces, J. Analyse Math. 51 (1988), 259–284. MR 963157, 10.1007/BF02791126
  • [Bu] 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
  • [Ch] E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
  • [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), no. 2-4, 235–247. MR 871732
  • [GL] Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, 𝐴_{𝑝} weights and unique continuation, Indiana Univ. Math. J. 35 (1986), no. 2, 245–268. MR 833393, 10.1512/iumj.1986.35.35015
  • [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. (2) 28 (1983), no. 1, 71–82. MR 703466, 10.1112/jlms/s2-28.1.71
  • [GT] 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
  • [Ha] Werner Haussmann, Approximation by harmonic functions, Topics in multivariate approximation (Santiago, 1986) Academic Press, Boston, MA, 1987, pp. 111–124. MR 924826
  • [HZ] Werner Haussmann and Karl Zeller, 𝐻-sets and best uniform approximation by solutions of elliptic differential equations, Results Math. 14 (1988), no. 1-2, 84–92. MR 956007, 10.1007/BF03323218
  • [HKL] 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
  • [He] 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
  • [HY] John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
  • [Hö] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
  • [In] E. L. Ince, Ordinary differential equations, Longmans, Green, London, 1927.
  • [KS] Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
  • [La] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
  • [Na1] Raghavan Narasimhan, Complex analysis in one variable, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781130
  • [Na2] Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. MR 0251745
  • [PW] Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
  • [SZ] Stanisław Saks and Antoni Zygmund, Analytic functions, Monografie Matematyczne, Tom XXVIII, Polskie Towarzystwo Matematyczne, Warszawa-Wroclaw, 1952. Translated by E. J. Scott. MR 0055432
  • [Si] Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Translated from the Romanian by Radu Georgescu. 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. MR 0270044
  • [Wi] J. Michael Wilson, A counterexample in the theory of best approximation, J. Approx. Theory 63 (1990), no. 3, 384–386. MR 1081038, 10.1016/0021-9045(90)90129-E
  • [Zw] D. Zwick, Uniform approximation by harmonic and subharmonic functions, Approximation theory VI, Vol. II (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 689–692. MR 1091082

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