Master's lecture in Mathematics - Bergur Snorrason

The lecture will be held via Zoom: https://eu01web.zoom.us/j/61656228257

Master's student: Bergur Snorrason

Title: Rudin-Carleson theorems

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Faculty:  Faculty of Physical Sciences

Advisor:  Benedikt Steinar Magnússon, Assistant Professor at the Faculty of Physical Sciences

Other members of the masters committee: Ragnar Sigurðsson, Professor at the Faculty of Physical Sciences

Examiner:  Tyson Ritter, Associate Professor at Universitetet i Stavanger

Abstract

The main theme of this thesis is the Rudin-Carleson extension theorem. It states a sufficient condition on a subset of the unit circle in C such that all continuous function thereon can be extended to a continuous function on the closed unit disk whose restriction to the open unit disk is holomorphic. The theorem is proved in two ways. Firstly in section 3.2, where it is done in the same manner as Rudin, and again in section 3.3, where it shown to be a corollary of a theorem of Bishop and the F. and M. Riesz theorem.

Section 3.1 is dedicated to proving the F. and M. Riesz theorem, which states a sufficient condition for a measure to be absolutely continuous with regards to the Lebesgue-measure on the unit circle. In section 3.3 we state and prove Bishop’s generalization of the Rudin-Carleson. We then show the Rudin-Carleson theorem to be a corollary of Bishop’s generalization and the F. and M. Riesz theorem. We look at the role of Bishop’s theorem in the classification of closed subsets of the boundary of the unit ball in C^n in section 4.1.

In section 4.2 we consider a way to relax the condition on the set in Bishop’s theorem. The set in Bishop’s theorem is conditioned such that all continuous functions thereon can be extended, that is, the condition is independent of the function we want to extend. The result of section 4.2 is a theorem that gives a sufficient condition on set such that a given continuous function thereon can be extended.