An introduction to harmonic analysis.
3rd ed.

*(English)*Zbl 1055.43001
Cambridge Mathematical Library. Cambridge: Cambridge University Press (ISBN 0-521-54359-2/pbk; 0-521-83829-0/hbk). xv, 314 p. (2004).

This is the third edition of the now classical textbook on harmonic analysis. As stated in the preface, the second edition was essentially identical to the first, except for the correction of a few misprints. In the current edition some more misprints were corrected, the wording changed in few places and some material added: two additional sections in Chapter I and one in Chapter IV; an additional appendix; and a few additional exercises. The book is divided into eight chapters.

Chapter I begins with a very rapid introduction to the basic concepts of Fourier series and convolution, and goes on to discuss such topics as norm and pointwise summability, the existence, for each convex positive sequence \(\{a_n\}\) decreasing to zero, of a Fourier cosine series with coefficients \(\{a_n\}\), Hilbert space and Fourier analysis in \(L^2(\mathbb T)\), absolutely convergent Fourier series, Fourier-Stieltjes coefficients, and, in the final set of exercises, distributions and pseudomeasures on \(\mathbb T\) and their Fourier coefficients. In Chapter II the author first relates the boundedness of the conjugation operator to norm convergence of Fourier series and then passes to the existence of continuous functions whose Fourier series diverge, the principle of localization, Dini’s test and set of divergence.

In Chapter III the author gives a short proof of the fact that the conjugation operator is of weak type \((1,1)\), proves the M. Riesz theorem, treats the Hardy-Littlewood maximal operator and its properties, and concludes with a discussion of the basic properties of \(H^p\) spaces. Chapter IV is devoted to the interpolation of operators and some of its consequences, such as the Hausdorff-Young theorem. In Chapter V the author discusses the good behaviour that results from the lacunarity of the Fourier series of a function, Sidon sets and the Denjoy-Carleman theorem on the quasi-analyticity of classes of infinitely differentiable functions.

Chapter VI is a long chapter on Fourier transforms on the line. Among the many topics in this chapter are (i) the systematic reduction of problems for the line to those for the circle (e.g., in the proofs of the M. Riesz theorem and Bochner’s theorem); (ii) tempered distributions, pseudomeasures, their transforms and their relation to the problem of spectral synthesis; (iii) an account of almost periodicity; (iv) the Paley-Wiener theorems relating \(L^2\) functions to their holomorphic extensions; (v) Kronecker’s theorem. Chapter VII is a brief sketch, indicating to the reader how the harmonic analysis developed to this point fits into the general picture of harmonic analysis on locally compact Abelian groups.

In the final chapter the author treats commutative Banach algebras and their applications to harmonic analysis. The applications cover such important topics as the Beurling-Helson theorem on automorphisms of \(L^1(\mathbb R)\), Wiener’s Tauberian theorem, Malliavin’s theorem on the failure of spectral synthesis, functions which “operate”, and the work of Varopoulos on tensor products and spectral synthesis.

Chapter I begins with a very rapid introduction to the basic concepts of Fourier series and convolution, and goes on to discuss such topics as norm and pointwise summability, the existence, for each convex positive sequence \(\{a_n\}\) decreasing to zero, of a Fourier cosine series with coefficients \(\{a_n\}\), Hilbert space and Fourier analysis in \(L^2(\mathbb T)\), absolutely convergent Fourier series, Fourier-Stieltjes coefficients, and, in the final set of exercises, distributions and pseudomeasures on \(\mathbb T\) and their Fourier coefficients. In Chapter II the author first relates the boundedness of the conjugation operator to norm convergence of Fourier series and then passes to the existence of continuous functions whose Fourier series diverge, the principle of localization, Dini’s test and set of divergence.

In Chapter III the author gives a short proof of the fact that the conjugation operator is of weak type \((1,1)\), proves the M. Riesz theorem, treats the Hardy-Littlewood maximal operator and its properties, and concludes with a discussion of the basic properties of \(H^p\) spaces. Chapter IV is devoted to the interpolation of operators and some of its consequences, such as the Hausdorff-Young theorem. In Chapter V the author discusses the good behaviour that results from the lacunarity of the Fourier series of a function, Sidon sets and the Denjoy-Carleman theorem on the quasi-analyticity of classes of infinitely differentiable functions.

Chapter VI is a long chapter on Fourier transforms on the line. Among the many topics in this chapter are (i) the systematic reduction of problems for the line to those for the circle (e.g., in the proofs of the M. Riesz theorem and Bochner’s theorem); (ii) tempered distributions, pseudomeasures, their transforms and their relation to the problem of spectral synthesis; (iii) an account of almost periodicity; (iv) the Paley-Wiener theorems relating \(L^2\) functions to their holomorphic extensions; (v) Kronecker’s theorem. Chapter VII is a brief sketch, indicating to the reader how the harmonic analysis developed to this point fits into the general picture of harmonic analysis on locally compact Abelian groups.

In the final chapter the author treats commutative Banach algebras and their applications to harmonic analysis. The applications cover such important topics as the Beurling-Helson theorem on automorphisms of \(L^1(\mathbb R)\), Wiener’s Tauberian theorem, Malliavin’s theorem on the failure of spectral synthesis, functions which “operate”, and the work of Varopoulos on tensor products and spectral synthesis.

Reviewer: Jean Ludwig (Metz)

##### MSC:

43-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis |

43A20 | \(L^1\)-algebras on groups, semigroups, etc. |

43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |

42A55 | Lacunary series of trigonometric and other functions; Riesz products |

43A40 | Character groups and dual objects |

42A20 | Convergence and absolute convergence of Fourier and trigonometric series |