Presenting new results along with research spanning five decades, Fractional Cauchy Transforms provides a full treatment of the topic, from its roots in classical complex analysis to its current state. Self-contained, it includes introductory material and classical results, such as those associated with complex-valued measures on the unit circle, that form the basis of the developments that follow. The authors focus on concrete analytic questions, with functional analysis providing the general framework.
After examining basic properties, the authors study integral means and relationships between the fractional Cauchy transforms and the Hardy and Dirichlet spaces. They then study radial and nontangential limits, followed by chapters devoted to multipliers, composition operators, and univalent functions. The final chapter gives an analytic characterization of the family of Cauchy transforms when considered as functions defined in the complement of the unit circle.
About the authors:
Rita A. Hibschweiler is a Professor in the Department of Mathematics and Statistics at the University of New Hampshire, Durham, USA.
Thomas H. MacGregor is Professor Emeritus, State University of New York at Albany and a Research Associate at Bowdoin College, Brunswick, Maine, USA.\
Definition of the families Fa
Relations between F1and H1
The Riesz-Herglotz formula
Representations with real measures and h1
The F. and M. Riesz theorem
The representing measures for functions in F1
The one-to-one correspondence between measures and functions in the Riesz-Herglotz formula
The Banach space structure of Fa
Norm convergence and convergence uniform on compact sets
BASIC PROPERTIES OF Fa o
Properties of the gamma function and the binomial coefficients
A product theorem
Membership of f and f ' in Fa
The inclusion of Fa in Fb when 0 = a