I. Functions of complex variables. 2. Mappings (Mathematics) 3. Fixed point theory. I. Title. QA33 1.7 . C37 1993 SIS'.9-dc20 92-32457

Representing measures and Jensen measures are studied for the uniform algebra on the infinite polydisk Δ¯∞ generated by the coordinate functions z1 z2,…. Let σ be the Haar measure on the infinite torus T∞, which is the distinguished boundary of the infinite polydisk. For fixed p in the range 1 < p, < ∞, it is shown that a point ζ ∈ Δ∞ has a representing measure in LP(σ) if and only if ζ ∈ l2. A related result for a class of representing measures for the origin, including the Haar measure σ, and for fixed p in the range 0 < p < ∞, is that the point evaluation at ζ is continuous in the Lp‐norm if and only if ζ ∈ Δ∞ ∩l2. In this case the functions in Hp are shown to correspond to analytic functions on the domain Δ∞ l2 in l2. Along these same lines, it is shown that H∞ (σ) is isometrically isomorphic to H (Δ∞ ∩ l2), and also to H∞(B), where B is the open unit ball of the sequence space c0.

These lecture notes are devoted to an area of current research interest that bridges functional analysis and function theory. The unifying theme is the notion of subharmonicity with respect to a uniform algebra. The topics covered include the rudiments of Choquet theory, various classes of representing measures, the duality between abstract sub-harmonic functions and Jensen measures, applications to problems of approximation of plurisubharmonic functions of several complex variables, and Cole's theory of estimates for conjugate functions. Many of the results are published here for the first time in monograph form.

Library of Congress Cataloging-in-Publication Data Gamelin, Theodore W.Complex analysis / Theodore W. Gamelin p. cm. -(Undergraduate texts in mathematics) Includes bibliographical references and index.

Abstract.We prove that the unit ball of a Banach space is polynomial-star dense in the unit ball of its bidual. This strengthens Goldstine's theorem on weak-star density.