On the spectra of computable bounded analytic functions

Abstract: McNicholl, in collaboration with Matheson and later individually, showed that a Blaschke product is computable if and only if it has a computable zero sequence with computable Blaschke constant. The spectrum of a Blaschke product is the set of accumulation points of its zeros. We use Matheson and McNicholl's results to consider the arithmetical complexity of such spectra for computable Blaschke products. Namely, we present results showing that all such spectra are $\Sigma^0_3$--closed, that there exists a $\Sigma^0_3$--complete spectrum, that every $\Pi^0_2$--closed subset of the unit circle is a spectrum, and that there exists a $\Sigma^0_2$--closed set which is not.

We then turn our attention to uniform Frostman Blaschke products, shown by Frostman to be those with nontangential limits of modulus one everywhere (as opposed to generic Blaschke products which, as inner functions, are only guaranteed to have radial limits of modulus one almost everywhere). Matheson showed that the spectra of such functions are precisely the closed, nonempty, and nowhere dense subsets of the unit circle. We discuss an effectivization of one direction of his result, showing that every computably closed, nonempty, and nowhere dense subset of the circle is the spectrum of a computable uniform Frostman Blaschke product.

Joint work with Timothy McNicholl

complex variableslogic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic