Loukas Grafakos

Classical Fourier Analysis (eBook, PDF)

• Format: PDF

Produktbeschreibung

The primary goal of these two volumes is to present the theoretical foundation of the field of Euclidean Harmonic analysis. The original edition was published as a single volume, but due to its size, scope, and the addition of new material, the second edition consists of two volumes. The present edition contains a new chapter on time-frequency analysis and the Carleson-Hunt theorem. The first volume contains the classical topics such as Interpolation, Fourier Series, the Fourier Transform, Maximal Functions, Singular Integrals, and Littlewood-Paley Theory. The second volume contains more recent topics such as Function Spaces, Atomic Decompositions, Singular Integrals of Nonconvolution Type, and Weighted Inequalities.

These volumes are mainly addressed to graduate students in mathematics and are designed for a two-course sequence on the subject with additional material included for reference. The prerequisites for the first volume are satisfactory completion of courses in realand complex variables. The second volume assumes material from the first. This book is intended to present the selected topics in depth and stimulate further study. Although the emphasis falls on real variable methods in Euclidean spaces, a chapter is devoted to the fundamentals of analysis on the torus. This material is included for historical reasons, as the genesis of Fourier analysis can be found in trigonometric expansions of periodic functions in several variables.

About the first edition:

"Grafakos's book is very user-friendly with numerous examples illustrating the definitions and ideas... 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."

- Kenneth Ross, MAA Online

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Produktdetails

• Verlag: Springer-Verlag GmbH
• Seitenzahl: 492
• Erscheinungstermin: 18. September 2008
• Englisch
• ISBN-13: 9780387094328
• Artikelnr.: 37288141

Autorenporträt

Loukas Grafakos is a Professor of Mathematics at the University of Missouri at Columbia.

Inhaltsangabe

Preface.- 1. Lp Spaces and Interpolation.- 2. Maximal Functions, Fourier Transform, and Distributions.- 3. Fourier Series.- 4. Topics on Fourier Series.- 5. Singular Integrals of Convolution Type.- 6. Littlewood-Paley Theory and Multipliers.- 7. Weighted Inequalities.- A. Gamma and Beta Functions.- B. Bessel Functions.- C. Rademacher Functions.- D. Spherical Coordinates.- E. Some Trigonometric Identities and Inequalities.- F. Summation by Parts.- G. Basic Functional Analysis.- H. The Minimax Lemma.- I. Taylor's and Mean Value Theorem in Several Variables.- J. The Whitney Decomposition of Open Sets in Rn.- Glossary.- References.- Index.

Rezensionen

"The most up-to-date account of the most important developments in the area. ... It has to be pointed out that the hard ones usually come with a good hint, which makes the book suitable for self-study, especially for more motivated students. That being said, the book provides a good reference point for seasoned researchers as well" (Atanas G. Stefanov, Mathematical Reviews, August, 2015)

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 Online

From 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)