Fractional Cauchy transforms

This monograph traces fractional Cauchy transforms from their origins in classical complex analysis to the present state of the field, and it mixes new findings with work accumulated over roughly fifty years. The early material lays the groundwork: introductory theory and classical results, including those for complex-valued measures on the unit circle, so readers can follow what comes later without outside sources. Throughout, the emphasis falls on specific analytic problems, while functional analysis supplies the broader structure in which they sit.

The central chapters move from basic properties to integral means and to the connections between these transforms and the Hardy and Dirichlet spaces. Next comes the study of radial and nontangential limits. Further chapters address multipliers, composition operators, and univalent functions. The closing chapter characterizes, in analytic terms, the Cauchy transforms regarded as functions on the complement of the unit circle.

The book is written by Rita A. Hibschweiler, a professor of mathematics and statistics at the University of New Hampshire in Durham, and Thomas H. MacGregor, professor emeritus at the State University of New York at Albany and research associate at Bowdoin College in Brunswick, Maine.