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 , (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.

**[1]**J. L. Walsh,*Interpolation and approximation by rational functions in the complex domain*, Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, R. I., 1935. MR**0218588 (36:1672b)****[2]**-,*On approximation to an analytic function by rational functions of best approximation*, Math. Z.**38**(1934), 163-176. MR**1545445****[3]**-,*On the overconvergence of certain sequences of rational functions of best approximation*, Acta Math.**57**(1931), 411-435. MR**1555339****[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