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On uniform convergence of rational, Newton-Padé interpolants of type (n, n) with free poles asn→∞

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Summary

Letf be meromorphic in ℂ. We show that there exists a sequence of distinct interpolation points {z j } ∞j=1 , and forn≧1, rational functions,R n (z) of type (n, n) solving the Newton-Padé (Hermite) interpolation problem,

$$R_n (z_j ) = f(z_j ), j = 1,2,...2n + 1,$$

and such that for each compact subsetK of ℂ omitting poles off, we have

$$\mathop {\lim }\limits_{n \to \infty } ||f - R_n ||_{L\infty (K)}^{1/n} = 0.$$

Extensions are presented to the case wheref(z) is meromorphic in a given open set with certain additional properties, and related results are discussed.

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References

  1. Baker, G.A.: Essentials of Padé approximants, 1st Ed. Academic Press, New York 1975

    Google Scholar 

  2. Baker, G.A., Graves-Morris, P.R.: Padé approximants, Part I: Basic theory, encyclopaedia of mathematics and its applications. Vol. 13. Reading: Addison Wesley 1981

    Google Scholar 

  3. Braess, D.: On the conjecture of Meinardus on rational approximation ofe x, II. J. Approx. Theory40, 375–379 (1984)

    Google Scholar 

  4. Buslaev, V.I., Goncar, A.A., Suetin, S.P.: On the convergence of subsequences of themth row of the Padé table. Math. USSR-Sb.48, 535–540 (1984)

    Google Scholar 

  5. Conte, S.D., de Boor, C.: Elementary numerical analysis, 2nd (student) Ed. McGraw Hill Kogakusha, Tokyo 1975

    Google Scholar 

  6. Goncar, A.A.: Some recent convergence results on diagonal Padé approximants. In: Chui, C.K., Schumaker, L., Ward, J.D. (eds.): Approximation theory V, pp. 55–70. New York: Academic Press 1986

    Google Scholar 

  7. Hayman, W.K., Meromorphic functions. 1st Ed. Oxford: Oxford University Press 1964

    Google Scholar 

  8. Hille, E.: Analytic function theory, 2nd Ed. New York: Chelsea 1973

    Google Scholar 

  9. Levin, A.L.: The distribution of poles of rational functions of best approximation and related questions, Math. USSR. Sb.9, 267–274 (1969)

    Google Scholar 

  10. Levin, A.L.: The distribution of the poles of the best approximating rational functions and the analytical properties of the approximated function. Israel. J. Math.24, 139–144 (1976)

    Google Scholar 

  11. Lubinsky, D.S.: Divergence of complex rational approximations. Pacific. J. Math.108, 141–153 (1983)

    Google Scholar 

  12. Lubinsky, D.S.: Padé tables of a class of entire functions. Proc. of Am. Math. Soc.94, 399–405 (1985)

    Google Scholar 

  13. Lubinsky, D.S.: Power series equivalent to rational functions: A shifting origin Kronecker type theorem, and normality of Padé tables. Numer. Math.54, 33–40 (1988)

    Google Scholar 

  14. Lubinsky, D.S., Saff, E.B.: Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials. Constructive Approximation3, 331–361 (1987)

    Google Scholar 

  15. Nuttall, J.: Asymptotics of diagonal Hermite-Padé polynomials. J. Approximation Theory42, 299–386 (1984)

    Google Scholar 

  16. Saff, E.B.: Polynomial and rational approximation in the complex domain. In: de Boor, C., (ed.): Approximation theory, pp. 21–49. Proc. Symp. Appl. Math., Vol. 36, Amer. Math. Soc., Providence 1986

    Google Scholar 

  17. Stahl, H.B.: Existence and uniqueness of rational interpolants with free and prescribed poles. In: Saff, E.B., (ed.): Approximation Theory, Tampa, pp. 180–209. Lect. Notes Math, Vol. 1287, Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  18. Stahl, H.B.: Three different approaches to a proof of convergence for Padé approximants. In: Gilewicz, J., Pindor, M., Siemaszko, W. (eds.): Rational approximation and its applications in mathematics and physics, pp. 79–124. Lect. Notes Math., Vol. 1237, Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  19. Trefethen, L.N.: The asymptotic accuracy of rational best approximation toe z on a disk. J. Approximation Theory40, 380–383 (1984)

    Google Scholar 

  20. Wallin, H.: The convergence of Padé approximants and the size of the power series coefficients. Appl. Anal.4, 235–251 (1975)

    Google Scholar 

  21. Walsh, J.L.: Interpolation and approximation by rational functions in the complex doman, 2nd Ed. Colloquium Publications, Vol. 20, Amer. Math. Soc., Providence 1956

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of the Witwatersrand, PO Wits 2050, Republic of South Africa

    D. S. Lubinsky

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  1. D. S. Lubinsky
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Deccember 26, 1988 to May 31, 1989: Department of Mathematics, University of South Florida, Tampa, PL, 33620, USA.

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Lubinsky, D.S. On uniform convergence of rational, Newton-Padé interpolants of type (n, n) with free poles asn→∞. Numer. Math. 55, 247–264 (1989). https://doi.org/10.1007/BF01390053

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