This monograph offers an introduction to finite Blaschke products and their connections to complex analysis, linear algebra, operator theory, matrix analysis, and other fields. Old favorites such as the Carathéodory approximation and the Pick interpolation theorems are featured, as are many topics that have never received a modern treatment, such as the Bohr radius and Ritt's theorem on decomposability. Deep connections to hyperbolic geometry are explored, as are the mapping properties, zeros, residues, and critical points of finite Blaschke products. In addition, model spaces, rational functions with real boundary values, spectral mapping properties of the numerical range, and the Darlington synthesis problem from electrical engineering are also covered.
Topics are carefully discussed, and numerous examples and illustrations highlight crucial ideas. While thorough explanations allow the reader to appreciate the beauty of the subject, relevant exercises following each chapter improve technical fluency with the material. With much of the material previously scattered throughout mathematical history, this book presents a cohesive, comprehensive and modern exposition accessible to undergraduate students, graduate students, and researchers who have familiarity with complex analysis.

Stephan Ramon Garcia is a professor of mathematics at Pomona College

Javad Mashreghi is a professor in the Department of Mathematics and Statistics at Laval University, Alexandre-Vachon. 

William T. Ross is a professor of mathematics and the Chair of the Department of Math and Computer Sciences at the University of Richmond.

Preface.- Notations.- 1. Geometry of the Schur class.- 2. Elementary hyperbolic geometry.- 3. Finite Blaschke products: the basics.- 4. Approximation by finite Blaschke products.- 5. Zeros and residues.- 6. Critical points.- 7. Interpolation.- 8. The Bohr radius.- 9. Finite Blaschke products and group theory.- 10. Finite Blaschke products and operator theory.- 11. Real complex functions.- 12. Finite-dimensional model spaces.- 13. The Darlington synthesis problem.- A. Some reminders.- Index.