##
**Harmonic analysis on semigroups. Theory of positive definite and related functions.**
*(English)*
Zbl 0619.43001

Graduate Texts in Mathematics, 100. New York etc.: Springer-Verlag. X, 289 p. DM 118.00; $ 44.10 (1984).

There are a number of simple ideas which are fundamental in harmonic analysis on locally compact topological groups and Lie groups which actually do not emerge clearly enough in the commutative case. One of these notions, that of a function of positive type, appears fortuitous and somewhat artificial until it is realized that it is the counterpart, for linear group representations, of the notion of a topologically cyclic (or monogenic) representation.

For instance, the recent solution of the radar synthesis problem (posed in 1953) is based on this connection and has been established by fibering the topologically irreducible, unitary, linear SchrĂ¶dinger representation of the real Heisenberg nilpotent Lie group over the dual of its center [cf. the reviewer, C. R. Math. Acad. Sci., Soc. R. Can. 6, 179-182 (1984; Zbl 0557.94003)]; ”Theta identities via Laguerre functions - a nilpotent harmonic analysis approach”, Rend. Semin. Mat., Torino (to appear; Zbl 0553.43004)].

Clearly, any group G is a semigroup with the natural involution \(x\mapsto x^*=x^{-1}\). In the case of a commutative semigroup \((G,+,*)\) with involution, this fact suggests defining a function \(f: G\to {\mathbb{C}}\) to be of positive type on G if and only if the matrix \((f(x^*_ j+x_ k))\), \(1\leq j\leq N\), \(1\leq k\leq N\), is positive definite for all choices of finite sequences \(x_ j\), \(1\leq j\leq N\), of elements in G.

Starting with a discussion of positive and negative definite kernels on sets, the main purpose of the monograph under review is to provide a thorough and self-contained treatment of functions of positive type on commutative semigroups with involution. Combined with a collection of closely related topics such as functions of negative type, completely monotone functions, Hoeffding’s inequality of probability theory and its generalizations, the authors show that there now exists a well- established and efficient theory called harmonic analysis on commutative semigroups. Unfortunately, no comparable theory seems to exist for noncommutative semigroups.

For instance, the recent solution of the radar synthesis problem (posed in 1953) is based on this connection and has been established by fibering the topologically irreducible, unitary, linear SchrĂ¶dinger representation of the real Heisenberg nilpotent Lie group over the dual of its center [cf. the reviewer, C. R. Math. Acad. Sci., Soc. R. Can. 6, 179-182 (1984; Zbl 0557.94003)]; ”Theta identities via Laguerre functions - a nilpotent harmonic analysis approach”, Rend. Semin. Mat., Torino (to appear; Zbl 0553.43004)].

Clearly, any group G is a semigroup with the natural involution \(x\mapsto x^*=x^{-1}\). In the case of a commutative semigroup \((G,+,*)\) with involution, this fact suggests defining a function \(f: G\to {\mathbb{C}}\) to be of positive type on G if and only if the matrix \((f(x^*_ j+x_ k))\), \(1\leq j\leq N\), \(1\leq k\leq N\), is positive definite for all choices of finite sequences \(x_ j\), \(1\leq j\leq N\), of elements in G.

Starting with a discussion of positive and negative definite kernels on sets, the main purpose of the monograph under review is to provide a thorough and self-contained treatment of functions of positive type on commutative semigroups with involution. Combined with a collection of closely related topics such as functions of negative type, completely monotone functions, Hoeffding’s inequality of probability theory and its generalizations, the authors show that there now exists a well- established and efficient theory called harmonic analysis on commutative semigroups. Unfortunately, no comparable theory seems to exist for noncommutative semigroups.

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

22A20 | Analysis on topological semigroups |

43A35 | Positive definite functions on groups, semigroups, etc. |

43A05 | Measures on groups and semigroups, etc. |