Hardy Spaces: (Series Number 179 Cambridge Studies in Advanced Mathematics)

The theory of Hardy spaces is a cornerstone of modern analysis. It combines techniques from functional analysis, the theory of analytic functions and Lesbesgue integration to create a powerful tool for many applications, pure and applied, from signal processing and Fourier analysis to maximum modulus principles and the Riemann zeta function. This book, aimed at beginning graduate students, introduces and develops the classical results on Hardy spaces and applies them to fundamental concrete problems in analysis. The results are illustrated with numerous solved exercises that also introduce subsidiary topics and recent developments. The reader's understanding of the current state of the field, as well as its history, are further aided by engaging accounts of important contributors and by the surveys of recent advances (with commented reference lists) that end each chapter. Such broad coverage makes this book the ideal source on Hardy spaces.

Table of Contents:
The origins of the subject; 1. The space H^2(T). An archetypal invariant subspace; 2. The H^p(D) classes. Canonical factorization and first applications; 3. The Smirnov class D and the maximum principle; 4. An introduction to weighted Fourier analysis; 5. Harmonic analysis and stationary filtering; 6. The Riemann hypothesis, dilations, and H^2 in the Hilbert multi-disk; Appendix A. Key notions of integration; Appendix B. Key notions of complex analysis; Appendix C. Key notions of Hilbert spaces; Appendix D. Key notions of Banach spaces; Appendix E. Key notions of linear operators; References; Notation; Index.

About the Author :
Nikolaï Nikolski is Professor Emeritus at the Université de Bordeaux working primarily in analysis and operator theory. He has been co-editor of four international journals and published numerous articles and research monographs. He has also supervised some thirty Ph.D. students, including three Salem Prize winners. Professor Nikolski was elected Fellow of the American Mathematical Society (AMS) in 2013 and received the Prix Ampère of the French Academy of Sciences in 2010.

Review :
'This is a beautiful introduction to a beautiful subject. The author does a masterful job of choosing topics that give a solid introduction to Hardy spaces without overwhelming the reader with too much too soon. Every student of mathematics should enjoy this book.' John McCarthy, Washington University, St Louis 'This is an excellent, highly-recommendable book by a leading expert in the field. Describing in a self-contained way the classical approach to Hardy spaces and their applications - with emphasis on the invariant subspace point of view - it also offers an entertaining journey into the history of twentieth-century analysis.' Joaquim Bruna, Universitat Autònoma de Barcelona 'The author has created a very interesting text that will serve as a source of basic material and an introduction for those interested in the interactions between harmonic analysis, complex analysis, and operator theory. Any researcher who masters the material in this text is well placed to read the more intensive monographs that further develop the interactions of these subjects.' Brett Wick, Washington University, St Louis 'From the high stand due to a lifelong affinity for the subject, Nikolaï Nikolski delineates and recounts the fascinating history of Hardy space with style. His inviting text is sparkled with biographical snapshots, colorful anecdotes, and insightful remarks, conveying a vivid picture of the field. The mathematical discourse is light, yet rigorous and self-contained. Carefully selected problems offer the earnest reader perspectives into current research. The apogee chapter is devoted to the immersion of the daunting Riemann hypothesis into Hardy space framework … Nikolski's text is a truly masterful piece of scholarship.' Mihai Putinar, University of California, Santa Barbara 'This is a comfortably paced introduction to the theory of Hardy spaces, starting at a level of advanced graduate students in analysis … Historical context and biographical details of the main researchers of the field is discussed more deeply than in comparable books.' M. Bona, Choice 'To help the reader through this material, Nikolski is both an experienced educator and writer and knows how to present the material, efficiently … so the student can learn as well as appreciate the subject. Nikolski also gives us plenty of historical vignettes of the main figures in the development of Hardy spaces and, especially for the student, gives several appendices for those needing some gentle reminders of measure theory, complex analysis, Hilbert spaces, Banach spaces, and operator theory.' William T. Ross, Bulletin of the American Mathematical Society