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Uniformly computable aspects of inner functions: estimation and factorization
Author(s) -
McNicholl Timothy H.
Publication year - 2008
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200710061
Subject(s) - mathematics , weierstrass factorization theorem , computability , factorization , bounded function , function (biology) , discrete mathematics , computability theory , computable function , pure mathematics , algebra over a field , mathematical analysis , algorithm , evolutionary biology , biology
The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of information necessary to compute the factorization of u . Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type‐Two Effectivity as our foundation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)