Axisymmetric harmonic interpolation polynomials in
Author:
Morris Marden
Journal:
Trans. Amer. Math. Soc. 196 (1974), 385402
MSC:
Primary 31B99
MathSciNet review:
0348130
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Abstract: Corresponding to a given function which is axisymnetric harmonic in an axisymmetric region and to a set of circles in an axisymmetric subregion , an axisymmetric harmonic polynomial is found which on the interpolates to or to its partial derivatives with respect to x. An axisymmetric subregion is found such that converges uniformly to on the closure of B. Also a is determined which, together with its first n partial derivatives with respect to x, coincides with on a single circle in and converges uniformly to in a closed torus with as central circle.
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 [1]
 S. Bergman, Integral operators in the theory of linear partial differential equations, Ergebnisse der Math. und ihrer Grenzgebiete, Heft 23, SpringerVerlag, Berlin, 1961. MR 25 #5277. MR 0141880 (25:5277)
 [2]
 P. J. Davis, Interpolation and approximation, Blaisdell, New York, 1963. MR 28 #393. MR 0157156 (28:393)
 [3]
 R. P. Gilbert, Function theoretic methods in partial differential equations, Math. in Science and Engineering, vol. 54, Academic Press, New York, 1969. MR 39 #3127. MR 0241789 (39:3127)
 [4]
 O. D. Kellogg, Foundations of potential theory, SpringerVerlag, Berlin, 1929. MR 0222317 (36:5369)
 [5]
 M. Marden, Value distributions of harmonic polynomials in several real variables, Trans. Amer. Math. Soc. 159 (1971), 137154. MR 43 #5046. MR 0279323 (43:5046)
 [6]
 L. M. MilneThomson, Theoretical hydrodynamics, 3rd ed., Macmillan, New York, 1956. MR 17, 796. MR 0075724 (17:796b)
 [7]
 G. Szegö, Orthogonal polynomials, 2nd rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR 21 #5029. MR 0106295 (21:5029)
 [8]
 V. I. Smirnov and N. A. Lebedev, The constructive theory of functions of a complex variable, ``Nauka", Moscow, 1964; English transl., Functions of a complex variable: Constructive theory, M.I.T. Press, Cambridge, Mass., 1968. MR 30 #2152; 37 #5369. MR 0171926 (30:2152)
 [9]
 J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 4th rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence R. I., 1965. MR 36 #1672b. MR 0218588 (36:1672b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403481306
PII:
S 00029947(1974)03481306
Keywords:
Axisymmetric harmonic polynomial,
axisymmetric harmonic function,
harmonic interpolation polynomial,
Bergman operator method
Article copyright:
© Copyright 1974
American Mathematical Society
