The Cauchy Transform
About this Title
Joseph A. Cima, University of North Carolina, Chapel Hill, Chapel Hill, NC, Alec L. Matheson, Lamar University, Beaumont, TX and William T. Ross, University of Richmond, Richmond, VA
Publication: Mathematical Surveys and Monographs
Publication Year 2006: Volume 125
ISBNs: 978-0-8218-3871-6 (print); 978-1-4704-1352-1 (online)
MathSciNet review: MR2215991
MSC: Primary 30-02; Secondary 30E10, 30E20, 46E15, 46E20, 46E22, 47B33, 47B38
The Cauchy transform of a measure on the circle is a subject of both classical and current interest with a sizable literature. This book is a thorough, well-documented, and readable survey of this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation theory as covered by Clark, Poltoratski, and Aleksandrov and contains an in-depth treatment of Clark measures.
Graduate students and research mathematicians interested in classical and modern complex analysis.
Table of Contents
- 1. Preliminaries
- 2. The Cauchy transform as a function
- 3. The Cauchy transform as an operator
- 4. Topologies on the space of Cauchy transforms
- 5. Which functions are Cauchy integrals?
- 6. Multipliers and divisors
- 7. The distribution function for Cauchy transforms
- 8. The backward shift on $H^2$
- 9. Clark measures
- 10. The normalized Cauchy transform
- 11. Other operators on the Cauchy transforms