The convergence of rational functions of best approximation to the exponential function. II
HTML articles powered by AMS MathViewer
- by E. B. Saff PDF
- Proc. Amer. Math. Soc. 32 (1972), 187-194 Request permission
Abstract:
Let ${W_{m,n}}(z)$ be a rational function of type (m, n) of best uniform approximation to the function ${e^z}$ on the closed unit disk. In this paper we show that any sequence $\{ {W_{m,n}}(z)\}$ for which $m + n \to \infty$ must converge to ${e^z}$ 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.References
-
M. Bôcher, Introduction to higher algebra, Macmillan, New York, 1907.
O. Perron, Die Lehre von den Kettencrüchen, 2nd ed., Chelsea, New York, 1929.
- E. B. Saff, On the row convergence of the Walsh array for meromorphic functions, Trans. Amer. Math. Soc. 146 (1969), 241–257. MR 265608, DOI 10.1090/S0002-9947-1969-0265608-2
- E. B. Saff, The convergence of rational functions of best approximation to the exponential function, Trans. Amer. Math. Soc. 153 (1971), 483–493. MR 274775, DOI 10.1090/S0002-9947-1971-0274775-5
- E. B. Saff, Regions of meromorphy determined by the degree of best rational approximation, Proc. Amer. Math. Soc. 29 (1971), 30–38. MR 281930, DOI 10.1090/S0002-9939-1971-0281930-2
- 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, DOI 10.1007/BF01170632
- 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 —, 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.
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 32 (1972), 187-194
- MSC: Primary 30A82
- DOI: https://doi.org/10.1090/S0002-9939-1972-0294656-7
- MathSciNet review: 0294656