Classical Fourier Analysis

The main goal of this text is to present the theoretical foundation of the field of Fourier analysis on Euclidean spaces. It covers classical topics such as interpolation, Fourier series, the Fourier transform, maximal functions, singular integrals, and LittlewoodPaley theory. The primary readership is intended to be graduate students in mathematics with the prerequisite including satisfactory completion of courses in real and complex variables. The coverage of topics and exposition style are designed to leave no gaps in understanding and stimulate further study.

This third edition includes new Sections 3.5, 4.4, 4.5 as well as a new chapter on Weighted Inequalities, which has been moved from GTM 250, 2nd Edition. Appendices I and B.9 are also new to this edition. Countless corrections and improvements have been made to the material from the second edition. Additions and improvements include: more examples and applications, new and more relevant hints for the existing exercises, new exercises, and improved references.

New edition extensively revised and updated, including 1000 different corrections and improvements in the existing text Includes a new chapter, "Topics on Fourier series", including sections on Gibbs phenomenon, summability methods and Jackson's theorem, Tauberian theorems, spherical Fourier inversion, and Fourier transforms on the line Provides motivation for the reader with more examples and applications, new and more relevant hints for the existing exercises, and about 20-30 new exercises in the existing chapters Includes supplementary material: sn.pub/extras Request lecturer material: sn.pub/lecturer-material

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

Inhalt

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. LittlewoodPaley 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.