Book review

This very interesting monograph presents the results of research done during the last three decades by the Bulgarian mathematician I.H. Dimovski and his coworkers on what they call convolutional calculus. Starting points are the Duhamel convolution integral and the operational calculus of J. Mikusinski. In a general setting, a convolution of a linear operator L is then defined as a bilinear, commutative and associative operation * in a linear space where L (f*g) = (Lf) * g holds for all elements f,g of this space. This generalizes the notion of the Duhamel integral. The book describes the systematic and successful search for new convolutions in important linear function spaces as well as their applications. Outside the scope of the volume are generalized convolution operations of probability theory and convolution products of the noncommutative stochastic calculus.

In a first chapter the foundations of the convolutional calculus are discussed including the Duhamel integral, the Mikusinski operator ring and convolution quotients, and convolutions of linear endomorphisms. The second chapter thoroughly describes convolutions of general integration operators and various applications, among others to Dirichlet expansions of locally holomorphic functions and to Bernoulli polynomials. Here a lot of interesting new and partly deep-lying results have been given. The final third chapter contains new convolutions connected with second-order linear differential operators, their initial value and boundary value problems and applications to partial differential equations of mathematical physics. Finite integral transformations of the Sturm-Liouville type and the Hankel type are also taken into consideration.

A list of 106 references with titles, an author index and a short subject index conclude the work. The text is lively written and supplemented with much background information, motivation and historical remarks. Definitions, lemmata, theorems and proofs are always followed by instructive examples and explanations. The volume is well produced. It can be recommended to all libraries and to research workers, teachers, and graduate students in the fields of mathematics, theoretical physics, systems theory and other engineering sciences.