Results concerning approximation to functions analytic on a closed point set R̄0 by arbitrary functions analytic and bounded in a region R1 containing R̄0 were first established by Walsh [4] in 1938 and later extended by the present writers to the limiting case where R̄0 and the boundary of R1 have points in common [5]. It is the purpose of the present note to continue the study of this problem now in situations where the approximated function is no longer assumed analytic at points common to the boundaries of R0 and R1.

This book is an account of the theory of Hardy spaces in one dimension, with emphasis on some of the exciting developments of the past two decades or so. The last seven of the ten chapters are devoted in the main to these recent developments. The motif of the theory of Hardy spaces is the interplay between real, complex, and abstract analysis. While paying proper attention to each of the three aspects, the author has underscored the effectiveness of the methods coming from real analysis, many of them developed as part of a program to extend the theory to Euclidean spaces, where the complex methods are not available.