On Closed Ideals of Analytic Functions

1. The closed ideals in the algebra A of all continuous functions fieie) on the unit circle X = {eie: 0^.B<2ir] which have analytic extensions/(z), \z\ <1 have been determined by Beurling and independently by Rudin [5] as follows: Let Hx denote the weak* closure [A]* of A as a subset of Laidm), where m denotes the normalized Lebesgue measure dd/2ir on the circle. A function qEHTM is called inner if \q\ =1 a.e. We shall regard two inner functions as the same if they are constant multiples of each other. It is well known [3, p. 66] that any such q can be extended to the open unit disc to be analytic there and the extended function is of the form