Approximation of Piecewise Continuous Functions by Quotients of Bounded Analytic Functions

Abstract

This paper concerns a certain subalgebra of the Banach algebra of complex valued, essentially bounded, Lebesgue measurable functions on the unit circle in the complex plane (denoted here by L∞). My interest in this subalgebra was prompted by a question of R. G. Douglas. Let H∞ denote the space of functions in L∞ whose Fourier coefficients with negative indices vanish (equivalently, the space of boundary functions for bounded analytic functions in the unit disk). Douglas [5] has asked whether every closed subalgebra of L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3]. At the time of this writing, no algebra is known for which the above question can be answered negatively.