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Modern Fourier Analysis

Loukas Grafakos
Publication Date: 
Number of Pages: 
Gratuate Texts in Mathematics 250
[Reviewed by
Michael Berg
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Grafakos’ Classical Fourier Analysis appears to have it all. It’s the first part of a double-barreled presentation on Fourier analysis, being a newly bifurcated version of a first edition called Classical and Modern Fourier Analysis. The book was so enthusiastically received as to call in short order for the present souped-up second edition, consisting of two books, one on Classical Fourier Analysis and a second on Modern Fourier Analysis. The author obviously vetted the books across a huge spectrum of Fourier- and complex analysts, as his “Acknowledgements” section indicates, and he 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.

What is particularly striking is how quickly Grafakos gets out of the starting blocks, with the first 160 pages of Classical Fourier Analysis traversing an orbit ranging from the (already serious) early business of interpolation in Lp spaces to a thorough discussion of distributions, convolution in Lp, and oscillatory integrals. From the outset this is hard analysis for hard analysts.

Grafakos then goes on to discuss, in a dense but thorough fashion, Fourier analysis on the (real) n-torus (addressing a host of deep theorems such as those concerning point wise convergence of Fourier series, convergence in norm, convergence presque partout) and singular integrals, including Calderón-Zygmund theory and vector valued singular integrals. And then it’s time for Littlewood-Paley theory, martingales, and even wavelets. Again: analysis with a vengeance!

The book also comes equipped with eleven appendices covering such topics as gamma- and beta-functions, Bessel functions, Rademacher functions, summation by parts, Schur’s lemma, and Whitney decomposition (replete with a cool picture: p. 463). These appendices are cut down to the bone, but are quite useful. (At this stage of my life, I might remember at least the statement of the Hahn-Banach theorem, but the Banach-Alaoglu theorem is another matter. Appendix G, “Basic Functional Analysis,” is a life-saver.)

Classical Fourier Analysis also comes equipped with a wealth of exercises (which look quite good, even to an outsider like me) and each chapter is capped off by a wonderful “Historical Notes” section testifying to the highest level of scholarship and the author’s keen attention to detail. The book is well-crafted in every way.

I haven’t had occasion to look at this kind of hard analysis since my senior year in college, three decades ago, and don’t qualify as anything other than a once-fellow-traveller with happy memories. But I feel entirely safe in noting that Classical Fourier Analysis is a terrific book, hitting the target at just the right level, and, granting that real effort is required form the reader or student (and that he should want to be a Fourier analyst when he grows up), the book should be tremendously successful in teaching fledglings the ropes. The hard work will pay off spectacularly!

And now we come to the second half of Grafakos opus, namely, Modern Fourier Analysis, for which the first book is truly a required prerequisite. As already indicated, the present second edition was spurred on by the enthusiasm readers showed for the first edition and it is clear that this success inspired Grafakos to turn up the heat. And, to be sure, Modern Fourier Analysis doesn’t let up in launching the willing student on a fast track to professional Fourier analysis, so to speak.

With the two books geared for a two-semester sequence, Modern Fourier Analysis starts off with the sixth chapter of the whole, titled, “Smoothness and function spaces,” and, indeed, we encounter some very young mathematical objects here, e.g. Sobolev spaces and Hardy spaces (well, maybe I should say, “middle-aged objects”); also, and properly so, singular integrals continue to do a lot of the heavy lifting here. I do recall that these players figured hugely in research seminars I attended in the late 1970s, underscoring that the qualifier “modern” is apposite.

This is brought out even more emphatically by what follows, namely, material on B(ounded) M(ean) O(scillation), i.e., spaces of functions possessing this property (recall Fefferman’s famous result: (Hardy)* = BMO. See p. 132 of Modern Fourier Analysis), singular integrals again (and forever, of course), weighted inequalities, and, as a crescendo at the end, the difficult but ever-so-important topic of the boundedness and convergence of Fourier integrals, followed by time-frequency analysis. All of this is serious, important, difficult, and elegant mathematics.

Just as Classical Fourier Analysis, its predecessor, Modern Fourier Analysis sports a huge number off well-designed problems and exercises, and every chapter ends with an exceedingly informative set of “Historical Notes.” However, Modern Fourier Analysis provides no appendices — I guess the earlier eleven are enough.

This two-volume set by Grafakos all but makes me wish for eight more hours in every day and a return of the manic energy of my hyperactive youth. Studying these books closely is bound to be a hugely enjoyable and eminently worthwhile experience; I think it’s nigh-on indispensable for the aspiring Fourier analyst.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.


Smoothness and Function Spaces

BMO and Carleson Measures

Singular Integrals of Nonconvolution Type

Weighted Inequalities

Boundedness and Convergence of Fourier Integrals

Time-Frequency Analysis and the Carleson-Hunt Theorem