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Convergence and applications of reproducing kernels for classes of discrete harmonic functions

Published online by Cambridge University Press:  17 April 2009

C. Wayne Mastin
C. Wayne Mastin
Affiliation:
Department of Mathematics, Mississippi State University, Mississippi, USA.
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Abstract

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This paper gives convergence properties and applications of the discrete analogs of reproducing kernels for various families of harmonic functions. From these results information is obtained on the solution of interpolation problems, the convergence of the discrete Neumann's function, and the solution to problems involving the discrete biharmonic operator.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 3 , December 1974 , pp. 339 - 358
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Aronszajn, N., “Theory of reproducing kernels”, Trans. Amer. Math. Soc. 68 (1950), 337–404.Google Scholar
[2]Bergman, Stefan, The kernel function and conformal mapping, 2nd revised ed. (Mathematical Surveys, 5. Amer. Math. Soc., Providence, Rhode Island, 1970).Google Scholar
[3]Chalmers, Bruce L., “Subspace kernels and minimum problems in Hilbert spaces with kernel function”, Pacific J. Math. 31 (1969), 619–628.CrossRefGoogle Scholar
[4]Courant, R., Friedrichs, K. und Lewy, H., “Über die partiellen Differenzengleichungen der mathematischen Physik”, Math. Ann. 100 (1928), 32–74.Google Scholar
[5]Deeter, Charles R. and Springer, George, “Discrete harmonic kernels”, J. Math. Mech. 14 (1965), 413–438.Google Scholar
[6]Epstein, Bernard, Partial differential equations. An introduction (International Series in Pure and Applied Mathematics. McGraw-Hill, New York, San Francisco, Toronto, London, 1962).Google Scholar
[7]Epstein, Bernard, Orthogonal families of analytic functions (Macmillan, New York; Collier-Macmillan, London; 1965).Google Scholar
[8]Forsythe, George E., Wasow, Wolfgang R., Finite-difference methods for partial differential equations (Applied Mathematics Series. John Wiley & Sons, New York, London, 1960).Google Scholar
[9]Huddleston, Robert E., “Maximum principle methods for the discretization error in the approximation of the Bergman harmonic kernel by finite difference methods”, SIAM J. Numer. Anal. 5 (1968), 613–625.CrossRefGoogle Scholar
[10]Laasonen, Pentti, “Über die erste und zweite Randwertaufgabe der präharmonischen und harmonischen Funktionen”, Ann. Acad. Sci. Fennicae. Ser. A.I. Math.-Phys. No. 40 (1948), 28pp.Google Scholar
[11]Mangad, Moshe, “Bounds for the two-dimensional discrete harmonic Green's function”, Math. Comp. 20 (1966), 60–67.Google Scholar
[12]McCrea, W.H. and Whipple, F.J.W., “Random paths in two and three dimensions”, Proc. Roy. Soc. Edinburgh 60 (1940), 281–298.Google Scholar
[13]Meschkowski, Herbert, Hilbertsche Räume mit Kernfunktion (Die Grundlehren der mathematischen Wissenschaften, 113. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1962).CrossRefGoogle Scholar
[14]Verblunsky, S., “Sur les fonctions préharmoniques”, Bull. Sci. Math. (2) 73 (1949), 148–152.Google Scholar
[15]Zaremba, S., “Sur le calcul numerique des fonctions demandées dans le problèms de Dirichlet et le problèms hydrodynamique”, Bull. Internat. Acad. Sci. Cracovie (1909), 125–195.Google Scholar