Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Some results concerning the boundary zero sets of general analytic functions


Author: Robert D. Berman
Journal: Trans. Amer. Math. Soc. 293 (1986), 827-836
MSC: Primary 30D40
DOI: https://doi.org/10.1090/S0002-9947-1986-0816329-6
MathSciNet review: 816329
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Two results concerning the boundary zero sets of analytic functions on the unit disk $ \Delta $ are proved. First we consider nonconstant analytic functions $ f$ on $ \Delta $ for which the radial limit function $ {f^{\ast}}$ is defined at each point of the unit circumference $ C$. We show that a subset $ E$ of $ C$ is the zero set of $ {f^{\ast}}$ for some such function $ f$ if and only if it is a $ {\mathcal{G}_\delta }$ that is not metrically dense in any open arc of $ C$. We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.


References [Enhancements On Off] (What's this?)

  • [1] N. U. Arakeljan, Uniform and tangential approximation with analytic functions, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 3 (1968), no. 45, 273-286. (Russian) MR 0274770 (43:530)
  • [2] M. G. Arsove, The Lusin-Privalov theorem for subharmonic functions, Proc. London Math. Soc. (3) 14 (1964), 260-270. MR 0160927 (28:4136)
  • [3] K. F. Barth and W. J. Schneider, On the impossibility of extending the Riesz uniquenss theorem to functions of slow growth, Ann. Acad. Sci. Fenn. AI 432 (1968), 1-9. MR 0247092 (40:361)
  • [4] -, An asymptotic analog of the F. and M. Riesz radial uniqueness theorem, Proc. Amer. Math. Soc. 22 (1969), 53-54. MR 0247095 (40:364)
  • [5] C. L. Belna, P. Colwell and G. Piranian, Radial limits and distribution by mechanical inheritance, preprint, Univ. of Michigan, Ann Arbor, 1984.
  • [6] R. D. Berman, Elementary proofs of some asymptotic radial uniquenss theorems, Proc. Amer. Math. Soc. 86 (1982), 226-229. MR 667279 (83k:30027)
  • [7] -, A converse to the Lusin-Privalov radial uniqueness theorem, Proc. Amer. Math. Soc. 87 (1983), 103-106. MR 677242 (84m:30048)
  • [8] -, The sets of fixed radial limit value for inner functions, Illinois J. Math. 29 (1985), 191-219. MR 784519 (86j:30046)
  • [9] -, Analogues of radial uniqueness theorems for subharmonic functions in the unit disk, J. London Math. Soc. (2) 29 (1984), 103-112. MR 734996 (85d:31003)
  • [10] -, A note on the Lusin-Privalov radial uniqueness theorem and its converse, Proc. Amer. Math. Soc. 92 (1984), 64-66. MR 749892 (85i:30061)
  • [11] A. Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1-13. MR 0001370 (1:226a)
  • [12] R. Cahill, On bounded functions satisfying averaging conditions, Trans. Amer. Math. Soc. 206 (1975), 163-174. MR 0367208 (51:3450)
  • [13] G. T. Cargo, Some topological analogues of the F. and M. Riesz uniqueness theorem, J. London Math. Soc. (2) 12 (1975), 67-74. MR 0390223 (52:11049)
  • [14] E. F. Collingwood and A. J. Lohwater, Theory of cluster sets, Cambridge Univ. Press, Cambridge, 1966. MR 0231999 (38:325)
  • [15] J. S. Hwang and P. Lappan, On a problem of Berman concerning radial limits, preprint, Michigan State Univ., 1984. MR 796466 (86m:30035)
  • [16] J. E. Littlewood, On functions subharmonic in a circle (II), Proc. London Math. Soc. 28 (1928), 383-394.
  • [17] N. N. Lusin and I. I. Privalov, Sur l'unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup. 42 (1925), 143-191.
  • [18] I. I. Privalov, Randeigenschaften analytischer Funktionen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. MR 0083565 (18:727f)
  • [19] F. Riesz and M. Riesz, Über die Randwerte einer analytischen Funktion, Quatrième Congrès des Math. Scand. Stockholm, 1916, pp. 27-44.
  • [20] P. J. Rippon, The boundary cluster sets of subharmonic functions, J. London Math. Soc. (2) 17 (1978), 469-479. MR 500632 (81h:30036)
  • [21] A. Samuelsson, On radial zeros of Blaschke products, Ark. Mat. 7 (1968), 477-494. MR 0241649 (39:2988)
  • [22] K. F. Tse, An analog of the Lusin-Privalov radial uniqueness theorem, Proc. Amer. Math. Soc. 25 (1970), 310-312. MR 0262509 (41:7115)
  • [23] Z. Zahorski, Über die Menge der Punkt in welchen die Ableitung unendlich ist, Tôhoku Math. J. 48 (1941), 321-330. MR 0027825 (10:359h)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30D40

Retrieve articles in all journals with MSC: 30D40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0816329-6
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society