Harmonic analysis on free groups. (English) Zbl 0536.43001

Figà-Talamanca, Alessandro; Picardello, Massimo A.

Lecture Notes in Pure and Applied Mathematics, Vol. 87. New York - Basel: Marcel Dekker, Inc. VIII, 145 p. SFr 94.00 (1983).

This book contains some aspects of harmonic analysis on free groups. Most of the contents are taken from fairly recent papers by the authors and others. After the introduction (chapter I) there are three parts:
1) Norm estimates for convolution operators on \(\ell^ 2(G)\) (chapter II, following U. Haagerup, and C. Akemann and P. Ostrand)
2) Spherical functions, radial functions, uniormly bounded representations, and related topics (chapters III-VI, this part is shaped to resemble the case of SL(2,\({\mathbb{R}})\). Part of chapter IV and chapter VI follow A. M. Mantero and A. Zappa).
3) Applications to local limit theorems, Fourier and Fourier-Stieltjes algebra, \(A_ p(G)\) and convolution operators on \(\ell^ p(G)\) (chapters VII-VIII).
Each chapter finishes with a section ”Notes and Remarks”, which is quite useful. It is a pity that there are quite a few misprints, omissions, and imprecisions. Nevertheless the book is interesting and nice to read.

Reviewer: M.Leinert

MSC:

43-02	Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
43A15	\(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A50	Convergence of Fourier series and of inverse transforms
43A55	Summability methods on groups, semigroups, etc.
43A65	Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)

Keywords:

free group; convolution operator; radial function; spherical function; uniformly bounded representation; Fourier algebra; Fourier-Stieltjes algebra; local limit theorem