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Cauchy transforms of characteristic functions and algebras generated by inner functions

Authors: Alec L. Matheson and Michael I. Stessin
Journal: Proc. Amer. Math. Soc. 133 (2005), 3361-3370
MSC (2000): Primary 46J10; Secondary 46J15, 30D50, 30D55
Published electronically: May 9, 2005
MathSciNet review: 2161161
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Abstract: We prove that Cauchy transforms of characteristic functions of subsets of positive measure of the unit circle are equidistributed in the unit disk in the sense that the $L^p$-closure of the polynomial algebra in these Cauchy transforms coincides with the $L^p$-closure of the polynomial algebra in a canonical inner function. As a corollary to this result we find conditions describing when the polynomial algebra in two singular inner functions determined by point masses is dense in the Hardy spaces $H^p$.

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  • 1. A. B. Aleksandrov, Measurable partitions of the circle induced by inner functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 149 (1986), Issled. Linein. Teor. Funktsii XV, 103-106, 188; translation in J. Soviet Math. 42 (1988), no. 2, 1610-1613. MR 0849298 (87i:30065)
  • 2. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math., 81 (1949), 239-255. MR 0027954 (10:381e)
  • 3. R. G. Blumenthal, Holomorphically closed algebras of analytic functions, Math. Scan. 34 (1974), 84-90. MR 0380423 (52:1323)
  • 4. J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. MR 0628971 (83g:30037)
  • 5. E. Hewitt, Remark on orthonormal sets in $L^2(a,b)$, Amer. Math. Monthly 61 (1954), 249-250. MR 0060146 (15:631e)
  • 6. Alexander Izzo, private communication.
  • 7. M. J. Jaffrey, T. L. Lance, M. I. Stessin, Submodules of the Hardy space over polynomial algebras, Pacific J. Math. 194 (2000), No. 2, 373-392. MR 1760788 (2001e:46097)
  • 8. J. Jones, Generators of the disk algebra, Ph.D. dissertation, Brown University, 1977.
  • 9. T. W. Körner, Fourier analysis, Cambridge University Press, Cambridge, 1988. MR 0924154 (89f:42001)
  • 10. W. Seidel, On the distribution of values of bounded analytic functions, Trans. Amer. Math. Soc. 36 (1934), 201-236. MR 1501738
  • 11. N. Sibony, J. Wermer, Generators for $A(\Omega )$, Trans. Amer. Math. Soc. 194 (1974), 103-114. MR 0419838 (54:7856)
  • 12. M. I. Stessin, P. J. Thomas, Algebras generated by two bounded holomorphic functions, J. d'Analyse Math. 90 (2003), 89-114. MR 2001066 (2004g:46070)
  • 13. J. Wermer, Rings of analytic functions, Ann. of Math. 67 (1958), 497-516. MR 0096817 (20:3299)
  • 14. J. Wermer, Subalgebras of the disk algebra, Colloque d'Analyse Harmonique et Complexe, Univ. Aix-Marseille I, Marseille, 1977, 7pp. (not consecutively paged). MR 0565008 (81e:46035)

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Additional Information

Alec L. Matheson
Affiliation: Department of Mathematics, Lamar University, Beaumont, Texas 77710

Michael I. Stessin
Affiliation: Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222

Received by editor(s): May 5, 2004
Received by editor(s) in revised form: June 28, 2004
Published electronically: May 9, 2005
Additional Notes: This work was accomplished while the first author was visiting the University at Albany. He thanks that institution for the hospitality extended during his visit.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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