##
**L’analyse harmonique. Son développement historique. (Harmonic analysis. Its historical development).**
*(French)*
Zbl 0695.43001

Paris etc.: Masson. x, 330 p. FF 370.00 (1990).

This book relates the history of harmonic analysis from the beginning of the theory of topological groups until recently. It takes place halfway between a book of mathematics and a work of historian. It describes especially the setting up of the main concepts and methods of this important branch of mathematics: topological group, Haar measure, Banach algebra, spectral synthesis, amenability.

The author begins by describing the emergence of the notion of group: group of substitutions in the Galois theory, transformation group in the work of Lie, the setting up of the axioms of topological groups. Then he explains how was solved the important problem of the invariant measure on a locally compact group, allowing the introduction of a Fourier transform on an arbitrary abelian locally compact group, and the junction of group theory and Fourier series. He relates the history of the representation theory since Frobenius and Schur, with the decisive step of the work of Peter and Weyl. Concerning recent developments, the author chooses some directions, among them the problem of spectral synthesis and the amenability.

This book is richly documented, and excessively instructive. The author does not content himself with relating the history of harmonic analysis, but very often he analyses the proofs in detail. On its own the bibliography with 900 references has a great value.

The author begins by describing the emergence of the notion of group: group of substitutions in the Galois theory, transformation group in the work of Lie, the setting up of the axioms of topological groups. Then he explains how was solved the important problem of the invariant measure on a locally compact group, allowing the introduction of a Fourier transform on an arbitrary abelian locally compact group, and the junction of group theory and Fourier series. He relates the history of the representation theory since Frobenius and Schur, with the decisive step of the work of Peter and Weyl. Concerning recent developments, the author chooses some directions, among them the problem of spectral synthesis and the amenability.

This book is richly documented, and excessively instructive. The author does not content himself with relating the history of harmonic analysis, but very often he analyses the proofs in detail. On its own the bibliography with 900 references has a great value.

Reviewer: J.Faraut

### MSC:

43-03 | History of abstract harmonic analysis |

22-03 | History of topological groups |

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A55 | History of mathematics in the 19th century |

01A60 | History of mathematics in the 20th century |

20-03 | History of group theory |