##
**Harmonic analysis on symmetric spaces and applications. I.**
*(English)*
Zbl 0574.10029

New York etc.: Springer-Verlag. xv, 341 p., 54 figs. DM 138.00 (1985).

A more descriptive title of the book under review might be: Teach yourself harmonic analysis on symmetric spaces by examples. The present Volume I deals with Fourier analysis, on \(\mathbb{R}^m\), harmonic analysis on the two-sphere and on the Poincaré upper half-plane. Volume II (to appear) will be devoted to harmonic analysis on the space of positive definite \(n\times n\) matrices and on the general noncompact symmetric space.

Chapter I (82 pp.) deals with standard material on Fourier analysis on \(\mathbb{R}^m\) (distributions, Fourier integral, Laplace integral, Fourier series, Poisson summation formula with applications to the theta-function, Mellin transform with applications to Epstein and Dedekind zeta-functions).

The brief second chapter (37 pp.) deals with harmonic analysis on the sphere as a typical example of a compact symmetric space. Topics covered include spherical harmonics with applications to quantum mechanics and the Radon transform.

The main part of the book is formed by Chapter III (176 pp.) on harmonic analysis on the Poincaré upper half-plane \(H\). First harmonic analysis on the full upper half-plane is dealt with in considerable detail with an extensive discussion of the various integral transforms and their inversion formulas. In the following sections the author covers harmonic analysis on \(\Gamma\setminus H\) where \(\Gamma\) is a discrete subgroup of \(\mathrm{SL}(2,\mathbb{R})\). The theory is developed for the modular group \(\Gamma = \mathrm{SL}(2,\mathbb{Z})\) but there are also hints to more general groups. There is a paragraph on classical modular forms (Hecke’s theory, theta functions) and some of their applications.

A substantial part of Chapter III is devoted to real-analytic automorphic forms (Maaß wave forms, non-holomorphic Eisenstein series). Maaß’ theory of Dirichlet series with functional equations associated with non- holomorphic automorphic forms is developed and the Hecke operators are discussed on spaces of Maaß wave forms. A highlight of the book is the last paragraph dealing with the Roelcke-Selberg spectral resolution of the hyperbolic Laplacian on \(\Gamma \setminus H\) and the Selberg trace formula and some of their applications.

The present book is not written in the familiar definition-theorem-proof-style. The level of presentation is informal and the details of most of the proofs are left as exercises to the reader. But the reader is supplied with numerous hints and a huge number of references to the literature that enable him (or her) to fill the gaps. The author emphasizes motivation and the common aspects of harmonic analysis on the various symmetric spaces under consideration. There are many concrete examples, some history and many hints on applications in mathematics, physics and engineering.

Altogether the author covers an enormous amount of material, and the numerous references will greatly help the reader to find his (or her) way through the jungle of the literature. (Non-American readers will appreciate to have an opportunity to update their language skills by learning recent official abbreviations.)

Chapter I (82 pp.) deals with standard material on Fourier analysis on \(\mathbb{R}^m\) (distributions, Fourier integral, Laplace integral, Fourier series, Poisson summation formula with applications to the theta-function, Mellin transform with applications to Epstein and Dedekind zeta-functions).

The brief second chapter (37 pp.) deals with harmonic analysis on the sphere as a typical example of a compact symmetric space. Topics covered include spherical harmonics with applications to quantum mechanics and the Radon transform.

The main part of the book is formed by Chapter III (176 pp.) on harmonic analysis on the Poincaré upper half-plane \(H\). First harmonic analysis on the full upper half-plane is dealt with in considerable detail with an extensive discussion of the various integral transforms and their inversion formulas. In the following sections the author covers harmonic analysis on \(\Gamma\setminus H\) where \(\Gamma\) is a discrete subgroup of \(\mathrm{SL}(2,\mathbb{R})\). The theory is developed for the modular group \(\Gamma = \mathrm{SL}(2,\mathbb{Z})\) but there are also hints to more general groups. There is a paragraph on classical modular forms (Hecke’s theory, theta functions) and some of their applications.

A substantial part of Chapter III is devoted to real-analytic automorphic forms (Maaß wave forms, non-holomorphic Eisenstein series). Maaß’ theory of Dirichlet series with functional equations associated with non- holomorphic automorphic forms is developed and the Hecke operators are discussed on spaces of Maaß wave forms. A highlight of the book is the last paragraph dealing with the Roelcke-Selberg spectral resolution of the hyperbolic Laplacian on \(\Gamma \setminus H\) and the Selberg trace formula and some of their applications.

The present book is not written in the familiar definition-theorem-proof-style. The level of presentation is informal and the details of most of the proofs are left as exercises to the reader. But the reader is supplied with numerous hints and a huge number of references to the literature that enable him (or her) to fill the gaps. The author emphasizes motivation and the common aspects of harmonic analysis on the various symmetric spaces under consideration. There are many concrete examples, some history and many hints on applications in mathematics, physics and engineering.

Altogether the author covers an enormous amount of material, and the numerous references will greatly help the reader to find his (or her) way through the jungle of the literature. (Non-American readers will appreciate to have an opportunity to update their language skills by learning recent official abbreviations.)

Reviewer: Jürgen Elstrodt (Münster)

### MSC:

11F03 | Modular and automorphic functions |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

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

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11M35 | Hurwitz and Lerch zeta functions |

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |

58C40 | Spectral theory; eigenvalue problems on manifolds |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

22E30 | Analysis on real and complex Lie groups |

43A85 | Harmonic analysis on homogeneous spaces |