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Padé approximation and gaussian quadrature

Published online by Cambridge University Press:  17 April 2009

G.D. Allen
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA.
C.K. Chui
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA.
W.R. Madych
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA.
F.J. Narcowich
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA.
P.W. Smith
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA.
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Abstract

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For certain types of formal power series, including the series of Stieltjes, we prove that the [n, n+j], j ≥ −1, Padé approximants coincide with certain gaussian quadrature formulae and hence, convergence of these approximants follows immediately.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Allen, G.D., Chui, C.K., Madych, W.R., Narcowich, F.J., and Smith, P.W., “Padé approximants and orthogonal polynomials”, Bull. Austral. Math. Soc. 10 (1974), 263270.CrossRefGoogle Scholar
[2]Allen, G.D., Chui, C.K., Madych, W.R., Narcowich, F.J., and Smith, P.W., “On the structure of the Padé table”, submitted.Google Scholar
[3]Allen, G.D., Chui, C.K., Madych, W.R., Narcowich, F.J., and Smith, P.W., “Padé approximation of Stieltjes series”, submitted.Google Scholar
[4]Baker, George A. Jr, “The theory and application of the Padé approximant method”, Advances in Theoretical Physics 1 (1965), 158.Google Scholar
[5]Franzen, N.R., “Convergence of Padé approximants for a certain class of meromorphic functions”, J. Approximation Theory 6 (1972), 264271.CrossRefGoogle Scholar
[6]Gragg, W.B., “The Padé table and its relation to certain algorithms of numerical analysis”, SIAM Rev. 14 (1972), 162.CrossRefGoogle Scholar
[7]Loeffel, J.J., Martin, A., and Simon, B. and Wightman, A.S., “Padé approximants and the anharmonic oscillator”, Phys. Lett. 30B (1969), 656658.CrossRefGoogle Scholar
[8]Pommerenke, Ch., “Padé approximants and convergence in capacity”, J. Math. Anal. Appl. 41 (1973), 775780.CrossRefGoogle Scholar
[9]Stroud, A.H., Secrest, Don, Gaussian quadrature formulae (Prentice-Hall, Englewood Cliffs, New Jersey, 1966).Google Scholar
[10]Wall, H.S., Analytic theory of continued fractions (Van Nostrand, New York, Toronto, London, 1948).Google Scholar
[11]Wallin, H., “The convergence of Padé approximants and the size of the power series coefficients”, Applicable Anal, (to appear).Google Scholar