Classical and Modern Fourier Analysis

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9780130353993: Classical and Modern Fourier Analysis

An ideal refresher or introduction to contemporary Fourier Analysis, this book starts from the beginning and assumes no specific background. Readers gain a solid foundation in basic concepts and rigorous mathematics through detailed, user-friendly explanations and worked-out examples, acquire deeper understanding by working through a variety of exercises, and broaden their applied perspective by reading about recent developments and advances in the subject. Features over 550 exercises with hints (ranging from simple calculations to challenging problems), illustrations, and a detailed proof of the Carleson-Hunt theorem on almost everywhere convergence of Fourier series and integrals of L p functions—one of the most difficult and celebrated theorems in Fourier Analysis. A complete Appendix contains a variety of miscellaneous formulae. L p Spaces and Interpolation. Maximal Functions, Fourier transforms, and Distributions. Fourier Analysis on the Torus. Singular Integrals of Convolution Type. Littlewood-Paley Theory and Multipliers. Smoothness and Function Spaces. BMO and Carleson Measures. Singular Integrals of Nonconvolution Type. Weighted Inequalities. Boundedness and Convergence of Fourier Integrals. For mathematicians interested in harmonic analysis.

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About the Author:

Loukas Grafakos is a native of Athens, Greece. He earned his doctoral degree at UCLA and is currently a Professor of Mathematics at the University of Missouri. He has taught at Yale University and Washington University in St. Louis and he has also held visiting positions at the Mathematical Sciences Research Institute in Berkeley and the University of Pittsburgh. He has been named a Kemper Fellow for Excellence in Teaching and he has authored or co-authored over forty research articles in Fourier analysis. An avid traveler, he has visited over one hundred countries and has given many international lectures.

Excerpt. © Reprinted by permission. All rights reserved.:

The word analysis comes from the Greek, which means "dissolving into pieces." This is usually the first step of a process that leads to a careful study and understanding of an object or phenomenon. The antithetical process is equally significant as it assembles the analyzed pieces after they have been individually examined. This procedure is the heart of Fourier analysis. Through its aorta, this heart disseminates information to a variety of applications. Fourier analysis is therefore a prism that diffracts ideas into a rainbow of uses and applications, making the subject one of the richest and most far-reaching in mathematics.

The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate students in mathematics and is designed to serve for a three-course sequence on the subject. The only prerequisite for understanding the text is satisfactory completion of a course in measure theory, Lebesgue integration, and complex variables. This book is intended to present the selected topics in some 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.

The choice of the material in the text reflects a measure of personal taste; however, a certain effort has been made to include a variety of topics of general interest. Much attention is given to details, which are designed to facilitate the understanding of first-time readers. Based on my personal experience, I felt a need to include details related to topics that articles often omit, leaving beginners to struggle through without explanation. Although it will behoove many readers to skim through the more technical aspects of the presentation and concentrate on the flow of ideas, the mere fact that details are here for reference will be comforting to some. I hope that students will profit from this comprehensive presentation and learn how to do mathematics rigorously. Unfortunately, including so many details has led to the large size of the book. But as one's maturity and familiarity with the subject increases, topics slowly become natural and reading is significantly accelerated.

The exercises that follow each section enrich the material of the corresponding section and provide an opportunity to develop additional intuition and deeper comprehension. Some of them are rather rudimentary and require minimal skill, while others are more interesting and challenging. Only a few exercises are considered difficult, but these are given with hints. A special effort has been made to prepare the exercises, which unfortunately did not double, but almost tripled, the amount of time and effort it took to complete this text. I hope that the reader will find this extra effort beneficial.

The historical notes given at the end of each chapter are intended to provide an accurate account of past research but also to suggest directions for further investigation. This book was partly written with the purpose of attracting students to research. Many of the topics in Chapter 10 lead to open problems that have bewildered mathematicians for decades. It is hoped that many students will be fascinated by the easy statements, yet the delicate complexity of some of these problems, and pursue a deeper understanding.

The text is completely self-contained as the appendix includes the miscellaneous material needed throughout. Certain user-friendly conventions have been adopted to facilitate searching. For instance, theorems, propositions, definitions, lemmas, remarks, and examples are numbered according to the order in which they appear in each section. Exercises are numbered similarly and can be easily located.

As this book is intended for a three-course sequence on the subject, I would like to suggest a slowly paced initial breakdown of the material, flexible enough to accommodate adjustments: Semester I: Chapters 1, 2, 3, and 4. Semester II: Chapters 5, 6, 7, and 9. Semester III: Chapters 8, 10, and other topics. Sections or subsections marked by a star would normally be omitted in a yearly course.

I am solely responsible for any misprints, mistakes, and historical omissions in this book. Please contact me directly (loukasC~math.missouri.edu) if you have any comments, suggestions, improvements, or corrections. Instructors are also welcome to contact me to obtain further hints on the existing exercises in the text. Suggestions for other exercises are also welcome. A list of current errata with acknowledgements will be kept at the following URL:

http://www.math.missouri.edu/~loukas/FourierAnalysis.html

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