## Book review

 Author(s) Dimovski, Ivan H. CONVOLUTIONAL CALCULUS Kluwer Academic Publishers 1990 S. Rogosin

This book is the second edition of the known for the specialists in Analysis and its applloations Bulgarian version of the monograph (published in 1982).

Convolutions are extremely attractive and useful tools of Mathematics. The concrete realizations of this prominent idea have been used in many problems (initially connected with some classes of differential and integral equations). Anyway till now there is no unified approach to construction of convolutions. Some of the most known are based on the special type of integral operator. So at the end of forties J. Mikusinski has oonstructed a convolutional calculus on the base of Volterra integral operator. The Mikusinski's approach is purely algebraic and uses as a main construction so called convolution quotients ring. This type of convolutional calculus have been then used for the solution of many problems. Nevertheless it become impractical when all the elements of a convolutional algebra are its divisors of zero.

The author has proposed around the middle of sixties a new modification of Mikusinski's approach, based not on the convolutional quotients ring but on the multiplier quotients one. It gives some advantages for the case of convolutional algebra with divisors of zero (but remain to be equivalent to Mikusinski's approach in the opposite case). This new (multiplier quotient) convolutional calculus is successfully developed in the book. Besides is presented a number of its applications to different problems for 1st and 2nd order differential operators.

The book consists of three chapters, author and subject indexes and a list of references covering the close to book's subject literature until the early eighties.

The first chapter "Convolutions of linear operators. Multipliers and multiplier quotients" is of preparatory character. Here is carefully developed the Mikusinski's convolutional calculus including all necessary algebraic and functional constructions. Besides it is presented the main problem of Mikusinski's approach - if one takes an arbitrary convolution of linear operator, then in general may exist a divisor of zero of this convolution. In this case one cannot use the convolution quotients which are the basic in Mikusinski's construction. It was I. Dimovski who has proposed to use instead of them so called multiplier quotients. In the ch.1 are given some basic facts of the theory of multipliers and multiplier quotients, in particular a definition and main properties of multiplier quotients ring of an annihilators-free convolutional algebra.

The main goal of the second chapter "Convolutions of general integration operators. Applications" is to investigate on the base of the results of previous chapter convolutions for an arbitrary right inverse operator of the differentiation operator in various functional spaces. So it is determined the algebraic ground for this study - here are systematically used the properties of the abstract right inverse operators for the given linear operator. The latter theory is highly developed due to some results of mathematicians from different countries (in particular of group of polish mathematicians. It is appeared that generating functions of right inverse operators for operator of differentiation is in many cases an entire function of exponential type. The latter properties form another part of this chapter investigations. Among the spaces considered is the space of locally holomorphic functions on some domain in complex plane. The above mentioned connection between right inverse operators and entire functions of exponential type makes possible to apply the convolutional approach to the Dirichlet expansions of locally holomorphic functions. So it is close to some results of A.F. Leontiev and his successors.

Besides on the consideration are right inverse operators (and convolutions connected with them) for such operators as the backward shift operator, Gelfond Leontiev differentiation operator. The last part of this chapter is devoted not to abstract development of the new convolutional approach but to special case of this theory based on so called Bernulli integration operator.

The third chapter "Convolutions connected with second order linear differential operators" is the main in sense of its applicability. First of all the author deals with the square of differentiation. He has constructed different types of right inverses for this operator and corresponding convolutions. It is generalized then for the case of arbitrary linear second-order differential operators. These constructions are applied first of all to non-singular second-order differential operators determined by Sturm-Liouville and more general boundary value conditions. On the investigation are also some special boundary conditions. The using technics is highly based on the properties of different integral transformations. At last are given some applications of the developed method to the solution of so called non-local boundary value problems.

The attractiveness of the book is in presentation of the method which combine purely algebraic constructions and classical results of mathematical analysis. This allowed the author not only present the modern state of this theory but also show how one can use the proposed methods in the special type of mathematical problems. The contents of the book is oriented on the applications of the obtained results to some new classes of differential and integral equations.

The monograph can be interesting for the specialists in different sciences which work or use convolutional methods in their investigations (mathematicians, physicists, engineers etc).