Ideals in Rings of Analytic Functions with Smooth Boundary Values

Source abstract

Let A denote the Banach algebra of functions analytic in the open unit disc D and continuous in . If f and its first m derivatives belong to A, then the boundary function f(e iθ ) belongs to C m (∂D) . The space A m of all such functions is a Banach algebra with the topology induced by C m (∂D). If all the derivatives of/ belong to A, then the boundary function belongs to C ∞ (∂D) , and the space A ∞ all such functions is a topological algebra with the topology induced by C ∞ (∂D) . In this paper we determine the structure of the closed ideals of A ∞ (Theorem 5.3). Beurling and Rudin (see e.g. [ 7 , pp. 82-89; 10 ]) have characterized the closed ideals of A , and their solution suggests a possible structure for the closed ideals of A ∞ .