This is the third edition of the famous book (first published in 1968 by John Wiley & Sons), which earned the author the 2002 Leroy P. Steele prize for mathematical exposition. During the last thirty-five years, the textbook has been used by generations of graduate students as a friendly introduction to the central ideas of harmonic analysis. These are demonstrated on classical problems in the theory of Fourier series on a circle (Chapters I-V) and the Fourier transform of functions, measures and classes of tempered distributions on a line (Chapter VI). Attention is permanently paid to preparation for generalizations to harmonic analysis on locally compact commutative groups and commutative Banach algebras, which are briefly investigated in Chapters VII and VIII. The spectrum of L∞-functions is described in Chapter VI, and spectral synthesis in regular Banach algebras later on in Chapter VIII. The author uses both real and complex methods simultaneously for studying either Hardy spaces in Chapter III, or interpolation of operators in Chapter IV. Typical features of the author's style are indications on how one dimensional methods and results can be used in other parts of analysis, e.g., to get a spectral theorem for unitary operators or to characterize function spaces (e.g., Sobolev or Besov spaces) by trigonometric approximations. For the third edition the author added some facts, which are closely related to the subject of the book. Since harmonic analysis belongs to the core of analysis, this carefully written book can be highly recommended to anybody who is interested in analysis. The reading of it has been and will be a pleasure both for students and experts.

Reviewer:

jmil