Loukas Grafakos is a native of Athens, Greece. 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.
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When I was young, the primary source book for Fourier analysis was Antoni Zygmund's two-volume classic Trigonometric Series , 2nd edition, published in A lot has happened in this subject since then, and many fine books have been published in the interim.
In this review, I will focus on two recently published books. The books by Grafakos and Torchinsky were published eighteen years apart, so it seems unfair to compare them. However, I want to give good guidance to potential students and other readers. Both books are encyclopedic sources of a large amount of important modern Fourier analysis, but my blunt advice to a budding analyst would be to study Grafakos's book and keep a copy of Torchinsky's book handy.
Both books have lots of good exercises, many with hints. Many of Torchinsky's hints are quite detailed. Torchinsky claims that his book is user-friendly, but I did not find it as friendly as Grafakos's book.
Most of the book is presented in the classical theorem-proof style without much explanation of strategies and tactics. The chapters end with historical notes that are good accounts of the very old history, though on page 23 he refers to work Fourier did in Later chapters deal with more recent history, especially results in the 's to 's.
Grafakos's book also has historical notes, but the focus is on more recent work. 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. Moreover, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises.
In fact, he indicates that they took nearly triple the time and effort of the rest of the book. The book includes useful appendices, which make the book self-contained.
The only exception is the Banach-Alaoglu Theorem, which is used on page but is missing from Appendix G. Grafakos's book includes a slick treatment of some classical "real variables. Sometimes it is better to start with Torchinsky to see the essential ideas before studying the corresponding topic in Grafakos.
For example, compare the T 1 theorem in Torchinsky page with that in Grafakos page There are many other topics in Grafakos. The major topics covered in Torchinsky, that are not in Grafakos, are harmonic and subharmonic functions chapter 7 and boundary-value problems on C 1 -domains chapter Grafakos's index is quite good, but I had a couple of difficulties. Neither quasi-Banach space nor quasi-norm appear in the index. In both cases c is a positive constant independent of the variable quantities A and B.
There is also a handy Index of Notation. Both books provide modern treatments; neither book has any exercises. Krantz's book covers many topics covered by the books under review. Folland's book focuses on locally compact groups and representation theory. Kenneth A. Ross ross math. He was President of the MAA during His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor.
His most recent work has been on Markov chains and random walks on finite groups and other algebraic systems. He is the author of the book Elementary analysis: the theory of calculus , now in 14th printing , co-author of Discrete Mathematics with Charles R. Skip to main content.
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Classical and Modern Fourier Analysis
The primary goal of this book is to present the theoretical foundation of the field of Euclidean Harmonic analysis. This book is Modern in that is contains more recent topics such as function spaces, atomic decompositions, singular integrals of nonconvolution type, and weighted inequalities. This book is mainly addressed to graduate students in mathematics. The prerequisites are satisfactory completion of courses in real and complex variables, and knowledge of classical Fourier analysis topics.
Modern Fourier Analysis
Loukas Grafakos" is a native of Athens, Greece. 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. Modern Fourier Analysis.