Classical Fourier Analysis

Preface.- Lp Spaces and Interpolation.- Maximal Functions, Fourier Transform, and Distributions.- Fourier Analysis on the Torus.- Singular Integrals of Convolution Type.- Littlewood-Paley Theory and Multipliers.- Gamma and Beta Functions.- Bessel Functions.- Rademacher Functions.- Spherical Coordinates.- Some Trigonometric Identities and Inequalities.- Summation by Parts.- Basic Functional Analysis.- The Minimax Lemma.- The Schur Lemma.- The Whitney Decomposition of Open Sets in Rn.- Smoothness and Vanishing Moments.- Glossary.- References.- Index.

From a reviews:“Grafakos’s book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Morever, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises.” - Ken Ross, MAA OnlineFrom the reviews of the second edition:"The author … has produced a very well-written, polished, and exciting graduate textbook which easily doubles as a reference book in a number of areas belonging to or touching on Fourier analysis. … Classical Fourier Analysis also comes equipped with a wealth of exercise … and each chapter is capped off by a wonderful ‘Historical Notes’ … . I think it’s nigh-on indispensable for the aspiring Fourier analyst." (Michael Berg, MAA Online, January, 2009)“Intended for graduate students who wish to study Fourier analysis. … also suitable for self-study. Proofs are provided in great detail. Each chapter is followed by historical notes with references, often including a discussion of further results. There are numerous exercises of varying difficulty, with hints and references provided for the harder ones. … certainly a valuable and useful addition to the existing literature and can serve as textbooks or as reference books. Students will especially appreciate the extensive collection of exercises.” (Andreas Seeger, Mathematical Reviews, Issue 2011 c)“This book is intended to present the selected topics in depth and to stimulate further study in Fourier analysis. … proofs are provided in great detail and a large amount of exercises of varying difficulty were carefully prepared by the author … . This book is very interesting and useful. It is not only a good textbook, but also an indispensable and valuable reference for researchers … . The readers will certainly benefit a lot from the detailed proofs and the numerous exercises.” (Yang Dachun, Zentralblatt MATH, Vol. 1220, 2011)