Solution of a convergence problem in the theory of $T$-fractions
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- by Rolf M. Hovstad PDF
- Proc. Amer. Math. Soc. 48 (1975), 337-343 Request permission
Abstract:
Let $f$ be a function, holomorphic in $|z| < R$, where $R > 1$, normalized by $f(0) = 1$, and satisfying a boundedness condition of the form $|f(z) - 1| < K$. It is proved that a certain modification of the Thron continued fraction expansion of $f$ converges to $f$ uniformly on any $|z| \leq r < R$.References
- W. J. Thron, Some properties of continued fractions $1+d_0z+K(z/(1+d_n z))$, Bull. Amer. Math. Soc. 54 (1948), 206–218. MR 24528, DOI 10.1090/S0002-9904-1948-08985-6 W. B. Jones, Contributions to the theory of Thron continued fractions, Ph.D. Thesis, Vanderbilt University, Nashville, Tenn., 1963.
- William B. Jones and W. J. Thorn, Further properties of $T$-fractions, Math. Ann. 166 (1966), 106–118. MR 200425, DOI 10.1007/BF01361442
- Haakon Waadeland, On $T$-fractions of functions holomorphic and bounded in a circular disc, Norske Vid. Selsk. Skr. (Trondheim) 1964 (1964), no. 8, 19. MR 177100
- Haakon Waadeland, A convergence property of certain $T$-fraction expansions, Norske Vid. Selsk. Skr. (Trondheim) 1966 (1966), no. 9, 22. MR 225978
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 48 (1975), 337-343
- MSC: Primary 30A22
- DOI: https://doi.org/10.1090/S0002-9939-1975-0364612-1
- MathSciNet review: 0364612