|
Cauchy transforms of characteristic functions and algebras generated by inner functions
Author(s):
Alec
L.
Matheson;
Michael
I.
Stessin
Journal:
Proc. Amer. Math. Soc.
133
(2005),
3361-3370.
MSC (2000):
Primary 46J10;
Secondary 46J15, 30D50, 30D55
Posted:
May 9, 2005
MathSciNet review:
2161161
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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  -closure of the polynomial algebra in these Cauchy transforms coincides with the  -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  .
References:
-
- 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
 , 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
 , 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)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
46J10,
46J15, 30D50, 30D55
Retrieve articles in all Journals with
MSC (2000):
46J10,
46J15, 30D50, 30D55
Additional Information:
Alec
L.
Matheson
Affiliation:
Department of Mathematics, Lamar University, Beaumont, Texas 77710
Email:
matheson@math.lamar.edu
Michael
I.
Stessin
Affiliation:
Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222
Email:
stessin@math.albany.edu
DOI:
10.1090/S0002-9939-05-07913-X
PII:
S 0002-9939(05)07913-X
Received by editor(s):
May 5, 2004
Received by editor(s) in revised form:
June 28, 2004
Posted:
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
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
