Vol. 31, No. 2, 1969

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Approximation by inner functions

Ronald George Douglas and Walter Rudin

Vol. 31 (1969), No. 2, 313–320

Let L(T) denote the complex Banach algebra of (equivalence classes of) bounded measurable functions on the unit circle T, relative to Lebesgue measure m. The norm f of an f in L(T) is the essential supremum of |f| on T. The collection of all bounded holomorphic functions in the open unit disc U forms a Banach algebra which can be identified (via radial limits) with the norm-closed subalgebra H of L(T).

A function f in L(T) is unimodular if |f| = 1 a.e., on T. The inner functions are the unimodular members of H. It is well known that they play an important role in the study of H.

The main result (Theorem 1) is that the set of quotients of inner functions is norm-dense in the set of unimodular functions in L(T). One consequence of this (Theorem 7) is that the set of radial limits of holomorphic functions of bounded characteristic in U is norm-dense in L(T). It is also shown (Theorem 3, 4) that the Gelfand transforms of the inner functions separate points on the Šilov boundary of H, and this is used to obtain a new proof (and generalization) of a theorem of D. J. Newman (Theorem 4).

Mathematical Subject Classification
Primary: 46.55
Secondary: 30.00
Received: 10 March 1969
Published: 1 November 1969
Ronald George Douglas
Walter Rudin