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

Saff, E. B.

doi:10.1090/S0002-9947-1971-0274775-5

The convergence of rational functions of best approximation to the exponential function
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Trans. Amer. Math. Soc. 153 (1971), 483-493 Request permission

Abstract:

The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function $a{e^{yz}}$ on the disk $|z| \leqq \rho$. It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for $a{e^{yz}}$ on $|z| \leqq \rho$ converge uniformly to $a{e^{yz}}$ on each bounded subset of the plane.

References

J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587
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
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
Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition, Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. Translated by Larry L. Schumaker. MR 0217482, DOI 10.1007/978-3-642-85643-3

Die Lehre von den Kettenbrüchen

Topological spaces

E. B. Saff, Approximation by rational and meromorphic functions having a bounded number of free poles, Trans. Amer. Math. Soc. 141 (1969), 79–92. MR 243082, DOI 10.1090/S0002-9947-1969-0243082-X
Thomas Muir, A treatise on the theory of determinants, Dover Publications, Inc., New York, 1960. Revised and enlarged by William H. Metzler. MR 0114826