Approximation by strongly annular solutions of functional equations
HTML articles powered by AMS MathViewer
- by R. Daquila PDF
- Proc. Amer. Math. Soc. 138 (2010), 2505-2511 Request permission
Abstract:
A major result of this paper is that the set of all functions $g(z)$ such that $g$ is strongly annular and is a solution of a Mahler type of functional equation given by $g(z)=q(z)g(z^p)$ where $p\ge 2$ is an integer and $q$ is a polynomial with $q(0)=1$ is a dense first category set in the set of all holomorphic functions on the open unit disk with the topology of almost uniform convergence. A second result is that strongly annular solutions of these types of functional equations are dense in the space of holomorphic functions with Maclaurin coefficients of $\pm 1$ with the same topology.References
- Paul-Georg Becker, Effective measures for algebraic independence of the values of Mahler type functions, Acta Arith. 58 (1991), no. 3, 239–250. MR 1121085, DOI 10.4064/aa-58-3-239-250
- D. D. Bonar, On annular functions, Mathematische Forschungsberichte, Band XXIV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1971. MR 0450560
- D. D. Bonar and F. W. Carroll, Annular functions form a residual set, J. Reine Angew. Math. 272 (1975), 23–24. MR 417428
- John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901, DOI 10.1007/978-1-4612-6313-5
- R. Daquila, Strongly annular solutions of Mahler’s functional equation, Complex Variables Theory Appl. 32 (1997), no. 1, 99–104. MR 1448483, DOI 10.1080/17476939708814982
- F. W. Carroll, Dan Eustice, and T. Figiel, The minimum modulus of polynomials with coefficients of modulus one, J. London Math. Soc. (2) 16 (1977), no. 1, 76–82. MR 480955, DOI 10.1112/jlms/s2-16.1.76
- Russell W. Howell, Annular functions in probability, Proc. Amer. Math. Soc. 52 (1975), 217–221. MR 374398, DOI 10.1090/S0002-9939-1975-0374398-2
Additional Information
- R. Daquila
- Affiliation: Department of Mathematics, Muskingum University, New Concord, Ohio 43762
- Email: rdaquila@muskingum.edu
- Received by editor(s): July 31, 2009
- Received by editor(s) in revised form: October 20, 2009, and November 5, 2009
- Published electronically: February 18, 2010
- Communicated by: Walter Van Assche
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2505-2511
- MSC (2010): Primary 30D10, 30B30, 30E10, 41A30
- DOI: https://doi.org/10.1090/S0002-9939-10-10278-0
- MathSciNet review: 2607880