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Computable analysis and Blaschke products
Author(s):
Alec
Matheson;
Timothy
H.
McNicholl
Journal:
Proc. Amer. Math. Soc.
136
(2008),
321-332.
MSC (2000):
Primary 03F60, 30D50
Posted:
October 16, 2007
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Abstract:
We show that if a Blaschke product defines a computable function, then it has a computable sequence of zeros in which the number of times each zero is repeated is its multiplicity. We then show that the converse is not true. We finally show that every computable, radial, interpolating sequence yields a computable Blaschke product.
References:
-
- 1.
- J. Bak, R. J. Newman, Complex Analysis, 1st ed. (Springer-Verlag, New York, 1982). MR 671250 (84b:30001)
- 2.
- J. Caldwell, M. B. Pour-El. On a simple definition of computable functions of a real variable- with applications to functions of a complex variable. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 1 - 19. MR 0366638 (51:2885)
- 3.
- J. Garnett, Interpolating Blaschke products generate
 . Pacific Journal of Mathematics, vol. 173 (1996), pp. 501 - 510. MR 1394402 (97f:30050) - 4.
- J. Garnett, Bounded Analytic Functions, 1st ed. (Academic Press, 1981). MR 628971 (83g:30037)
- 5.
- P. Hertling, An effective Riemann Mapping Theorem. Theoretical Computer Science, vol. 219 (1999), pp. 225 - 265. MR 1694433 (2000f:03187)
- 6.
- G. A. Hjelle, Unimodular functions and interpolating Blaschke products. Proceedings of the American Mathematical Society, vol. 134 (2006), pp. 207 - 214. MR 2170560 (2006f:30039)
- 7.
- I. Kalantari and L. Welch, Recursive and nonextendible functions over the reals; filter foundation for recursive analysis. II. Annals of Pure and Applied Logic, vol. 98 (1999), no. 1 - 3, pp. 87 - 110. MR 1696843 (2000g:03098)
- 8.
- I. Kalantari and L. Welch, Specker's theorem, cluster points, and computable quantum functions, Logic in Tehran, Lecture Notes in Logic, vol. 26 (Association for Symbolic Logic, 2006). MR 2262318
- 9.
- A. Matheson and M. Stessin, Cauchy Transforms of characteristic functions and algebras generated by inner functions. Proceedings of the American Mathematical Society, vol. 133 (2005), pp. 3361 - 3370. MR 2161161 (2006f:30040)
- 10.
- W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill, 1987). MR 924157 (88k:00002)
- 11.
- R. I. Soare, Recursively Enumerable Sets and Degrees, 1st ed. (Springer-Verlag, Berlin, Heidelberg, 1987). MR 882921 (88m:03003)
- 12.
- E. Specker, Nicht konstruktiv beweisbare Sätze der Analysis. The Journal of Symbolic Logic, vol. 14 (1949), pp. 145 - 158. MR 0031447 (11:151g)
- 13.
- K. Weihrauch, Computable Analysis. An Introduction, 1st ed. (Springer-Verlag, Berlin, 2000). MR 1795407 (2002b:03129)
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Additional Information:
Alec
Matheson
Affiliation:
Department of Mathematics, Lamar University, Beaumont, Texas 77710
Timothy
H.
McNicholl
Affiliation:
Department of Mathematics, Lamar University, Beaumont, Texas 77710
Email:
mcnicholl@math.lamar.edu
DOI:
10.1090/S0002-9939-07-09102-2
PII:
S 0002-9939(07)09102-2
Received by editor(s):
June 2, 2006
Received by editor(s) in revised form:
January 27, 2007
Posted:
October 16, 2007
Dedicated:
Dedicated to the memory of Alec Matheson.
Communicated by:
Julia Knight
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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