By William B. Jones, W. J. Thron, Haakon Waadeland (eds.)

ISBN-10: 3540115676

ISBN-13: 9783540115670

ISBN-10: 3540392769

ISBN-13: 9783540392767

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**Extra resources for Analytic Theory of Continued Fractions: Proceedings of a Seminar-Workshop held at Loen, Norway, 1981**

**Example text**

Quelques recherches sur la theorie des quadratures mechanlques, Ann. Sci. Ec. , Paris (3) i (1884), 409-426. 20. , Orthogonal Polynomials, Math. , New York (1959). 21. , New York (1953). 22. , Convergence of infinite exponentials with complex elements, Proc. Amer. Math. Soc. 8 (1957), 1040-1043. 23. J. and Waadeland, Haakon, Survey of modifications fractions, these Lecture Notes. R. Acad. Sol. Paris 200 quadrature formulas, Math. G. Teubner, Stuttgart Colloquium Publications dltes vol. 23, Amer.

A new definition important. 42] ap~ications known as well as certain other arguments was for all sequences with interested in the That Hamel somehow did not seem to be of modifications that are now below, to what Hamel had to say. b 0 + K(an/b n) {Wn } fractions, When one of us fractions. that we shall mention closer attention fraction continued in the field. around 1960 he was mainly approach However with the various involving from workers be called convergent Wn E V([an},{bn}) 9 if it now seems His proposal {Sn(Wn)} The dependence of V on {a } and {b } was not specified.

Leads to an increase in the region of The first result is that lim Sn(Z ) = F(z) uniformly on any compact subset of Izl < ~ (under the same conditions as in the I I 2 paper [12]). The paper contains an improvement to Izl < ~ R , and in 1975 Hovstad 3 [13] extended it to any R' < R , again matched with the proper strength of the boundedness condition. See also [37]. 4) and the Lemmas I and 2 in [35] that (z) + 0 - Sn(0 ) and as n + ~ in Izl < i . I I 57 So actually there are two benefits of the transition from Sn(O) to Sn(Z) : acceleration of convergence and extension of region of convergence.

### Analytic Theory of Continued Fractions: Proceedings of a Seminar-Workshop held at Loen, Norway, 1981 by William B. Jones, W. J. Thron, Haakon Waadeland (eds.)

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