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Transactions of the American Mathematical Society

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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, 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

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