The convergence of rational functions of best approximation to the exponential function. II

Author:
E. B. Saff

Journal:
Proc. Amer. Math. Soc. **32** (1972), 187-194

MSC:
Primary 30A82

MathSciNet review:
0294656

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Abstract: Let be a rational function of type (*m, n*) of best uniform approximation to the function on the closed unit disk. In this paper we show that any sequence for which must converge to for all values of *z*. This is the first result which describes completely the regions of convergence of arbitrary sequences formed from a Walsh array.

**[1]**M. Bôcher,*Introduction to higher algebra*, Macmillan, New York, 1907.**[2]**O. Perron,*Die Lehre von den Kettencrüchen*, 2nd ed., Chelsea, New York, 1929.**[3]**E. B. Saff,*On the row convergence of the Walsh array for meromorphic functions.*, Trans. Amer. Math. Soc.**146**(1969), 241–257. MR**0265608**, 10.1090/S0002-9947-1969-0265608-2**[4]**E. B. Saff,*The convergence of rational functions of best approximation to the exponential function*, Trans. Amer. Math. Soc.**153**(1971), 483–493. MR**0274775**, 10.1090/S0002-9947-1971-0274775-5**[5]**E. B. Saff,*Regions of meromorphy determined by the degree of best rational approximation*, Proc. Amer. Math. Soc.**29**(1971), 30–38. MR**0281930**, 10.1090/S0002-9939-1971-0281930-2**[6]**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**, 10.1007/BF01170632**[7]**J. L. Walsh,*The convergence of sequences of rational functions of best approximation with some free poles*, Approximation of Functions (Proc. Sympos. General Motors Res. Lab., 1964 ), Elsevier Publ. Co., Amsterdam, 1965, pp. 1–16. MR**0186986****[8]**-,*Interpolation and approximation by rational functions in the complex domain*, 5th ed., Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, R.I., 1969.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0294656-7

Keywords:
Best uniform approximation,
rational functions of type (*m, n*),
Walsh array,
Padé table,
Bernstein lemma,
Taylor development,
polynomial of least squares approximation

Article copyright:
© Copyright 1972
American Mathematical Society