Abstract
The theory of bounded analytic functions is reexamined from the viewpoint of computability theory.
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References
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)
Boone, W.W.: The word problem. Proc. Natl. Acad. Sci. USA 44, 1061–1065 (1958)
Braverman, M., Cook, S.: Computing over the reals: foundations for scientific computing. Not. Am. Math. Soc. 53(3), 318–329 (2006)
Barry Cooper, S.: Computability Theory. Chapman & Hall/CRC, Boca Raton (2004)
Cutland, N.: Computability: An Introduction to Recursive Function Theory. Cambridge University Press, Cambridge (1980)
Frostman, O.: Potential d’equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddelanden Lunds Univ. Mat. Sem. 3, 1–118 (1935)
Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981)
Grzegorczyk, A.: On the definitions of computable real continuous functions. Fundam. Math. 44, 61–71 (1957)
Hertling, P.: An effective Riemann mapping theorem. Theor. Comput. Sci. 219, 225–265 (1999)
Kalantari, I., Welch, L.: Point-free topological spaces, functions and recursive points; filter foundation for recursive analysis. I. Ann. Pure Appl. Log. 93(1–3), 125–151 (1998)
Kalantari, I., Welch, L.: Recursive and nonextendible functions over the reals; filter foundation for recursive analysis. II. Ann. Pure Appl. Log. 98(1–3), 87–110 (1999)
Kleene, S.: On notation for ordinal numbers. J. Symb. Log. 3(4), 150–155 (1938)
Lerman, M.: Degrees of Unsolvability: Local and Global Theory. Perspectives in Mathematical Logic. Springer, Berlin (1983)
Matheson, A., McNicholl, T.H.: Computable analysis and Blaschke products. Proc. Am. Math. Soc. 136(1), 321–332 (2008)
McNicholl, T.H.: Uniformly computable aspects of inner functions: estimation and factorization. Math. Log. Q. 54(5), 508–518 (2008)
Myhill, J.: A recursive function defined on a compact interval and having a continuous derivative that is not recursive. Michigan J. Math. 18, 97–98 (1971)
Naftalevič, A.G.: On interpolation by functions of bounded characteristic. Vilniaus Valst. Univ. Moksl. Darb. Mat. Fiz. Chem. Moksl. Ser. 5, 5–27 (1956)
Odifreddi, P.G.: Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, 1st edn. North-Holland, Amsterdam (1989)
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989)
Pour-El, M.B., Caldwell, J.: On a simple definition of computable function of a real variable—with applications to functions of a complex variable. Z. Math. Log. Grundl. Math. 21, 1–19 (1975)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)
Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symb. Log. 14, 145–158 (1949)
Specker, E.: Der Satz vom Maximum in der rekursiven Analysis. In: Heyting, A. (ed.) Constructivity in Mathematics: Proceedings of the Colloquium Held at Amsterdam, 1957. Studies in Logic and the Foundations of Mathematics, pp. 254–265. North-Holland, Amsterdam (1959)
Turing, A.M.: On computable numbers, with an application to the entscheidungsproblem. Proc. Lond. Math. Soc., Ser. 2 42, 220–265 (1936)
Andreev, V.V., McNicholl, T.H.: Computing interpolating sequences. Theory Comput. Syst. 46(2), 340–350 (2010)
Weihrauch, K.: Computable Analysis. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2000)
Acknowledgements
I first of all want to thank my departed colleague Alec Matheson for introducing me to the beautiful world of bounded analytic functions and for setting in motion the research direction described in this paper. Many thanks to the referee and to Barry Cooper for their helpful comments. I also thank Javad Mashreghi for his encouragement. Finally, I thank my wife Susan for her support.
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McNicholl, T.H. (2013). On the Computable Theory of Bounded Analytic Functions. In: Mashreghi, J., Fricain, E. (eds) Blaschke Products and Their Applications. Fields Institute Communications, vol 65. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5341-3_13
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