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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some examples in degree of approximation by rational functions


Authors: D. Aharonov and J. L. Walsh
Journal: Trans. Amer. Math. Soc. 159 (1971), 427-444
MSC: Primary 30.70
DOI: https://doi.org/10.1090/S0002-9947-1971-0289787-5
MathSciNet review: 0289787
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Abstract: We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of $ E$, (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.


References [Enhancements On Off] (What's this?)

  • [1] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Third edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587
    J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. MR 0218588
  • [2] J. L. Walsh, On approximation to an analytic function by rational functions of best approximation, Math. Z. 38 (1934), no. 1, 163–176. MR 1545445, https://doi.org/10.1007/BF01170632
  • [3] J. L. Walsh, On the overconvergence of certain sequences of rational functions of best approximation, Acta Math. 57 (1931), no. 1, 411–435. MR 1555339, https://doi.org/10.1007/BF02403051
  • [4] O. Perron, Die Lehre von den Kettenbrüchen, Chelsea, New York, 1929.
  • [5] H. S. Wall, Analytic theory of continued fractions, Chelsea, New York, 1967.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0289787-5
Keywords: Approximation, rational functions, Padé table, best approximation, overconvergence
Article copyright: © Copyright 1971 American Mathematical Society