Algebras generated by the disc algebra and bounded harmonic functions
HTML articles powered by AMS MathViewer
- by Alexander J. Izzo PDF
- Proc. Amer. Math. Soc. 135 (2007), 1065-1071 Request permission
Abstract:
Let $D$ denote the open unit disc, and let $A(D)$ denote the disc algebra. The subsets $E$ of $\partial D$ such that the inclusion $A(D)[f,{\overline f}]\supset C(\overline D)$ holds for every nonconstant $f\in H^\infty (D)$ continuous on $E$, or the inclusion $A(D)[f] \supset C(\overline D)$ holds for every bounded harmonic nonholomorphic function $f$ on $D$ continuous on $E$, are characterized. In the first case the condition is that $E$ has positive measure, and in the second case that $E$ has full measure in $\partial D$.References
- A. B. Aleksandrov, Measurable partitions of the circle induced by inner functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (1986), no. Issled. Lineĭn. Teor. Funktsiĭ. XV, 103–106, 188 (Russian, with English summary); English transl., J. Soviet Math. 42 (1988), no. 2, 1610–1613. MR 849298, DOI 10.1007/BF01665047
- Herbert Alexander and John Wermer, Several complex variables and Banach algebras, 3rd ed., Graduate Texts in Mathematics, vol. 35, Springer-Verlag, New York, 1998. MR 1482798
- Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), no. 3, 631–638. MR 905614, DOI 10.1512/iumj.1987.36.36034
- E. M. Čirka, Approximation by holomorphic functions on smooth manifolds in $\textbf {C}^{n}$, Mat. Sb. (N.S.) 78 (120) (1969), 101–123 (Russian). MR 0239121
- Jacqueline Détraz, Algèbres de fonctions analytiques dans le disque, Ann. Sci. École Norm. Sup. (4) 3 (1970), 313–352. MR 435421
- A. M. Davie and A. Stray, Interpolation sets for analytic functions, Pacific J. Math. 42 (1972), 33–37. MR 313517
- P. Fatou, Series trigonométriques et séries de Taylor, Acta Math., vol. 30, 1906.
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- W. K. Hayman, Identity theorems for functions of bounded characteristic, J. London Math. Soc. (2) 58 (1998), no. 1, 127–140. MR 1666094, DOI 10.1112/S0024610798006334
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- E. A. Heard and J. H. Wells, An interpolation problem for subalgebras of $H^{\infty }$, Pacific J. Math. 28 (1969), 543–553. MR 243359
- Alexander J. Izzo, Algebras containing bounded holomorphic functions, Indiana Univ. Math. J. 52 (2003), no. 5, 1305–1342. MR 2010729, DOI 10.1512/iumj.2003.52.2315
- Alexander J. Izzo, Algebras generated by holomorphic and harmonic functions on the disc, Bull. London Math. Soc. 37 (2005), no. 5, 761–770. MR 2164839, DOI 10.1112/S0024609305004352
- Alexander J. Izzo, Some algebras of bounded functions on the disc, C. R. Math. Acad. Sci. Soc. R. Can. 27 (2005), no. 3, 72–75 (English, with English and French summaries). MR 2155659
- Alec L. Matheson and Michael I. Stessin, Cauchy transforms of characteristic functions and algebras generated by inner functions, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3361–3370. MR 2161161, DOI 10.1090/S0002-9939-05-07913-X
Additional Information
- Alexander J. Izzo
- Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
- MR Author ID: 307587
- Email: aizzo@math.bgsu.edu
- Received by editor(s): January 12, 2005
- Received by editor(s) in revised form: November 1, 2005
- Published electronically: September 26, 2006
- Communicated by: Juha M. Heinonen
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 1065-1071
- MSC (2000): Primary 46J10, 46J15, 30H05
- DOI: https://doi.org/10.1090/S0002-9939-06-08547-9
- MathSciNet review: 2262907