Analytic functions with prescribed cluster sets
HTML articles powered by AMS MathViewer
- by L. W. Brinn PDF
- Trans. Amer. Math. Soc. 300 (1987), 681-693 Request permission
Abstract:
Suppose that $0 < R \leq + \infty$. A monotonic boundary path (mb-path) in $\left | z \right | < R$ is a simple continuous curve $z = z(s)$, $0 \leq s < 1$, in $\left | z \right | < R$ such that $\left | {z(s)} \right | \to R$ strictly monotonically as $s \to 1$. Suppose that $f$ is a complex valued function, defined in $\left | z \right | < R$, and that $t$ is any mb-path in $\left | z \right | < R$. The cluster set of $f$ on $t$ is the set of those points $w$ on the Riemann sphere for which there exists a sequence $\{ {z_n}\}$ of points of $t$ with ${\operatorname {lim}_{n \to \infty }}\left | {{z_n}} \right | = R$ and ${\operatorname {lim}_{n \to \infty }}f({z_n}) = w$. The cluster set is denoted by ${C_t}(f)$. If the cluster set is a single point set, that point is called the asymptotic value of $f$ on $t$. If the function $f$ is continuous, then ${C_t}(f)$ is a continuum on the Riemann sphere. It is a conjecture of F. Bagemihl and W. Seidel that if $\mathcal {T}$ is a family of mb-paths in $\left | z \right | < R$ satisfying certain conditions, and if $\mathcal {K}$ is an analytic set of continua on the Riemann sphere, then there exists a function $f$, analytic in $\left | z \right | < R$, such that $\left \{ {{C_t}(f)|t \in \mathcal {T}} \right \} = \mathcal {K}$. A restricted form of the conjecture is mentioned in [3, p. 100]. Our principal results show the correctness of the conjecture in the case that $\mathcal {K}$ is the collection of all continua on the Riemann sphere and $\mathcal {T}$ is a tress of a certain type. The results are generalized in several directions. In particular, our technique for constructing the analytic function $f$ extends immediately to the case in which $\mathcal {K}$ is any closed set of continua on the sphere. Specific examples of closed sets lead to several corollaries.References
- N. U. Arakeljan, Uniform and tangential approximations by analytic functions, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 3 (1968), no. 4-5, 273–286 (Russian, with Armenian and English summaries). MR 0274770
- F. Bagemihl and J. E. McMillan, Uniform approach to cluster sets of arbitrary functions in a disk, Acta Math. Acad. Sci. Hungar. 17 (1966), 411–418. MR 203032, DOI 10.1007/BF01894884
- F. Bagemihl and W. Seidel, A problem concerning cluster sets of analytic functions, Math. Z. 62 (1955), 99–110. MR 72220, DOI 10.1007/BF01180626
- F. Bagemihl and W. Seidel, Some boundary properties of analytic functions, Math. Z. 61 (1954), 186–199. MR 65643, DOI 10.1007/BF01181342
- F. Bagemihl and W. Seidel, Spiral and other asymptotic paths, and paths of complete indetermination, of analytic and meromorphic functions, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 1251–1258. MR 59352, DOI 10.1073/pnas.39.12.1251
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- P. Gauthier and W. Seidel, Some applications of Arakélian’s approximation theorems to the theory of cluster sets, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 6, 458–464 (English, with Armenian and Russian summaries). MR 0302919 S. Kierst and E. Szpilrajn, Sur certaines singularités des fonctions analytiques uniformes, Fund. Math. 21 (1933), 276-294.
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751 —, Topology. II, Academic Press, New York, 1968. S. N. Mergelyan, Uniform approximation to functions of a complex variable, Uspekhi Mat. Nauk (N.S.) 7 (1952), 33-122; Amer. Math. Soc. Transl. 101 (1954).
- Alice Roth, Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen, Comment. Math. Helv. 11 (1938), no. 1, 77–125 (German). MR 1509593, DOI 10.1007/BF01199693
- J. L. Stebbins, A construction of meromorphic functions with prescribed asymptotic behavior, Nagoya Math. J. 41 (1971), 75–87. MR 291468
- 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
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 300 (1987), 681-693
- MSC: Primary 30D40; Secondary 30E10
- DOI: https://doi.org/10.1090/S0002-9947-1987-0876472-3
- MathSciNet review: 876472