Abstract
This is a survey of some basic ideas in the convergence theory for continued fractions, in particular value sets, general convergence and the use of modified approximants to obtain convergence acceleration and analytic continuation. The purpose is to show how these ideas apply to some other areas of mathematics. In particular, we introduce {w k }-modifications and general convergence for sequences of Padé approximants.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker, G.: Defects and the convergence of Padé approximants, Acta Appl. Math. 61 (2000), 37–52 (this issue).
Baker, I. N. and Rippon, P. J.: Towers of exponents and other composite maps, Complex Variables 12 (1989), 181–200.
Beardon, A. F.: On the convergence of Padé approximants, J. Math. Anal. Appl. 21 (1968), 344–346.
Chisholm, J. S. R.: Approximation by sequences of Padé approximants in regions of meromorphy, J. Math. Phys. 7 (1966), 39–44.
Denjoy, A.: Sur l'itération des fonctions analytiques, C.R. Acad. Sci. Paris 182 (1926), 255–257.
Jacobsen, L.: General convergence for continued fractions, Trans. Amer. Math. Soc. 294 (1986), 477–485.
Jacobsen, L.: Nearness of continued fractions, Math. Scand. 60 (1987), 129–147.
Jones, W. B. and Thron, W. J.: Continued fractions. Analytic theory and applications, Encyclopedia of Mathematics and Its Applications 11, Addison-Wesley, Reading, Mass, 1980. Now distributed by Cambridge University Press, New York.
Lorentzen, L.: Analytic continuation of functions given by continued fractions, revisited, Rocky Mountain J. Math. 23 (1993), 683–706.
Lorentzen, L.: Computation of limit periodic continued fractions. A survey, Numer. Algorithms 10 (1995), 69–111.
Lorentzen, L.: Convergence of compositions of self-mappings, Ann. Univ. Marie Curie Sk?odowska Sect. A 53(13) (1999), 121–145.
Lorentzen, L. and Ruscheweyh, St.: Simple convergence sets for continued fractions K.an=1/, J. Math. Anal. Appl. 179 (1993), 349–370.
Lorentzen, L. and Waadeland, H.: Continued Fractions with Applications, Studies in Comput. Math. 3, North-Holland, Amsterdam, 1992.
Perron, O.: Die Lehre von den Kettenbrüchen, 2. Band, 3. Aufl. Teubner, Stuttgart, 1957.
Scott, W. T. and Wall, H. S.: A convergence theorem for continued fractions, Trans. Amer. Math. Soc. 47 (1940), 155–172.
Scott, W. T. and Wall, H. S.: Value regions for continued fractions, Bull. Amer. Math. Soc. 47 (1941), 580–585.
Sylvester, J. J.: Note on a new continued fraction applicable to the quadrature of the circle, Philos. Magazine London 84 (1869), 373–375.
Thron,W. J. and Waadeland, H.: Accelerating convergence of limit periodic continued fractions K.an=1/, Numer. Math. 34 (1980), 155–170.
Thron, W. J. and Waadeland, H.: Convergence questions for limit periodic continued fractions, Rocky Mountain J. Math. 11 (1981), 641–657.
Wolff, J.: Sur l'itération des fonctions, C.R. Acad. Sci. Paris 182 (1926), 200–201.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lorentzen, L. Ideas from Continued Fraction Theory Extended to Padé Approximation and Generalized Iteration. Acta Applicandae Mathematicae 61, 185–206 (2000). https://doi.org/10.1023/A:1006414501573
Issue Date:
DOI: https://doi.org/10.1023/A:1006414501573