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Softcover ISBN:  9780821848647 
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Softcover ISBN:  9780821848647 
Product Code:  COLL/54.2.S 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470432003 
Product Code:  COLL/54.2.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Softcover ISBN:  9780821848647 
eBook ISBN:  9781470432003 
Product Code:  COLL/54.2.S.B 
List Price:  $188.00 $143.50 
MAA Member Price:  $169.20 $129.15 
AMS Member Price:  $150.40 $114.80 

Book DetailsColloquium PublicationsVolume: 54; 2005; 578 ppMSC: Primary 42; 05; 34; Secondary 47; 30
This twopart volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of onedimensional Schrödinger operators.
Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by \(z\) (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.
The book is suitable for graduate students and researchers interested in analysis.
ReadershipGraduate students and research mathematicians interested in analysis.
This item is also available as part of a set: 
Table of Contents

Chapters

Chapter 9. Rakhmanov’s theorem and related issues

Chapter 10. Techniques of spectral analysis

Chapter 11. Periodic Verblunsky coefficients

Chapter 12. Spectral analysis of specific classes of Verblunsky coefficients

Chapter 13. The connection to Jacobi matrices

Appendix A. Reader’s guide: Topics and formulae

Appendix B. Perspectives

Appendix C. Twelve great papers

Appendix D. Conjectures and open questions


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This twopart volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of onedimensional Schrödinger operators.
Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by \(z\) (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.
The book is suitable for graduate students and researchers interested in analysis.
Graduate students and research mathematicians interested in analysis.

Chapters

Chapter 9. Rakhmanov’s theorem and related issues

Chapter 10. Techniques of spectral analysis

Chapter 11. Periodic Verblunsky coefficients

Chapter 12. Spectral analysis of specific classes of Verblunsky coefficients

Chapter 13. The connection to Jacobi matrices

Appendix A. Reader’s guide: Topics and formulae

Appendix B. Perspectives

Appendix C. Twelve great papers

Appendix D. Conjectures and open questions