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Book review

Author(s) Aizenberg, Lev
Publisher Kluwer Academic Publishers
Year of publication 1993
Reviewed by S.Rogosin

The big part of this book was published originally in Russian under the title "Carleman's Formulas in Complex Analysis. First Applications" (Novosibirsk, Nauka. Siberian Branch, 1990). But in comparison with latter it is completely remade in accordance to modern results and approaches. Besides the last part of it (one third of the whole book) is new one - it is written especially for this edition.

The main idea of the book is to use the old style Carleman's formulas for complex analytic functions, to obtain their different generalizations (in one- or multi-dimensional Complex Analysis). as well as to give some applications of these formulas to various problems of mathematics, physics, signal processing etc. Roughly speaking Carleman-type formulas answer the following question: what are the values of analytic function (of one or several variables) in a domain, if there is some information about its values on some part of the boundary? This question is of nature of analytic functions. So, any success in this problem's solution give us immediately some results in different questions using the analytic functions techniques. There are several approaches to the solution of this problem. One of the widely spread is so called Carleman's approach goes back to the original Carleman's result of 1926. It is utilized in many works until now.

The author has successfully applied the Carleman-type methods of the above mentioned problems investigation. He has collected a number of formulas of such a type, discussed several generalizations of the main Carleman idea and his machinery. L.Aizenberg is working in this direction more than thirty years, he has initiated the successful work with these problems of many mathematicians (especially in Krasnoyarsk, Siberia). So this book looks like an encyclopedia on the theory and applications of the Carleman-type ideas and methods (the first attempt in this branch of mathematics).

The monograph consists of four parts divided into eleven chapters, a long list of references, historical and bibliographical notes and indexes.

The first part consists of two chapters (ch.1 "One-Dimensional Carleman Formulas" and ch.2 "Generalization of One-Dimensional Carleman Formulas") and is devoted to the description of some classical and modern results for analytic functions of one complex variable. First of all here is presented the original Carleman's approach given firstly for a special type domain on the complex plane. Developing the Carleman's idea G.N.Goluzin and V.I.Krylov have constructed the Carleman-type formula for an arbitrary simply connected domain in C. At last more than twenty years later M.M.Lavrentyev has generalized their approach for practically any one-dimensional situation. These methods are on the discussion in the first chapter. They are also basic for the constructing of some new formulas presented in the second chapter. The most of these results are obtained by the author, his cooperators and another representatives of the soviet complex analytic school.

The central part of the book is the second one (consisting of four chapters). On the discussion here are Carleman's formulas in multidimensional complex analysis. Except of a small number of classical results the most of others were received quite recently and for the first time presented in the monographic literature.

The chapter III ("Integral Representations of Holomorphic Functions of Several Complex Variables and Logarithmic Residues") has a preliminary character. Here are collected the basic necessary facts playing the main role in the construction of multidimensional analogs of Carleman's formulas.

The next three Chapters present a number of results of Carleman-type in C^n, First of all are considered the case when an analytic in a domain <formula> function can be rebuilt due to its known values on a subset of boundary with maximal dimension (ch. IV "Multidimensional Analogs of Carleman Formulas with Integration over Boundary sets of Maximal Dimension"). The most of results here are obtained in spirit of initial Carleman's idea (consisting in the construction of a "quenching" function allowed to overchange an integration on the whole boundary by integration over its part in integral representation formulas). In fact this idea does not work in C^n in many cases because of stronger than in C^1 property of analytic continuation. Anyway the Carleman-type formulas can be obtained for sections of the domain D by complex lines. The results of such a type are presented in the Chapter IV. They are based on the different integral representation formulas for analytic functions of several complex variables.

It was noticed that the uniqueness set for analytic functions of several complex variables could be "very small". On the base of this fact some Carleman-type formulas were obtained in the case of determining sets of smaller dimension. They are included in the chapter V ("Multidimensional Carleman Formulas for Sets of Smaller Dimension"). It was appeared that the contents of these formulas much more close to the initial Carleman idea than of many others.

At last in the ch. VI ("Carleman Formulas with in Homogeneous Domains") are collected some formulas for the special types of domains (for classical ones as balls and polydisks as well as for so called Siegel domains playing an important role in characterizing of homogeneous bounded domains).

Three chapters (VII - IX) are included in the part III "FIRST APPLICATIONS". Among the applications of previously mentioned results are those of two types - "interior" and "exterior" ones. The applications of the first kind deal with classical and modern problems for analytic functions of one or several complex variables. They are presented in the ch. VII ("Applications in Complex Analysis"). The most of them are connected with the property of analytic continuation. The applications of the second kind are those in physics and signal processing (ch. VIII). First of all the "quenching" function is introduced in the forward dispersion relations. Then are considered the problems of extrapolation type. On their base it is obtained an analog of the Kotelnikov's theorem allowed to get a number of new results in the communication theory. For some of constructed formulas are then conducted computing experiments. The latter are presented in the ch. IX.

The author has prepared an additional (in respect to Russian version of the book) part IV ("SUPPLEMENT TO THE ENGLISH EDITION"). It consists of two Chapters (ch. X "Criteria for Analytic Continuation. Harmonic Extension" and ch. XI "Carleman Formulas and Related problems"). Here are presented the most of results in the area obtained very recently.

The book is written in the extremely unified manner very close to that of the old Russian manuscripts. It is devoted to the specialists in complex analysis and its applications and to all using the complex analytic functions techniques in their research. The book is the encyclopedia-type collection of the results in the considered direction of mathematics. It has a wide list of references on subject. The involving language is very simple and clear but all the results have strict and careful proofs. All this made the book extremely useful for young mathematicians and for those having not too much knowledge in the area, but trying to apply some of these results in their work.