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The moment problem. (English) Zbl 1383.44004
Graduate Texts in Mathematics 277. Cham: Springer (ISBN 978-3-319-64545-2/hbk; 978-3-319-64546-9/ebook). xii, 535 p. (2017).
It was the well-known mathematician P. L. Chebyshev who firstly posed, in 1873, a problem related to a moment problem, and who gave a partial solution of this problem. In fact, Chebyshev was mainly interested to obtain a limit theorem in probability theory. The investigations of Chebyshev in this direction were continued by A. Markov, who exhibited several solutions and generalizations of it. The term moment problem, that is, a precise definition of it, was firstly given by T. Stieljes in a memoir in two parts, dedicated to the study of continuous fractions, published in 1894 and 1895. He wrote the following: “We shall give the name moment problem to the following problem: it is required to find the distribution of positive mass on the interval $$[0,\infty)$$, given the moments of order $$k$$ $$(k=1,2,3,\dots)$$ of the distribution.” By such a distribution Stieltjes understood a non-decreasing function on $$[0,\infty)$$, say $$\sigma$$, whose generalized moments were defined as $$s_k=\int_0^\infty t^kd\sigma(t)$$, $$k=0,1,2,\dots$$, where $$s_0$$ is the total mass, $$s_1$$ is the static moment, and $$s_2$$ is the moment of inertia with respect to the point $$t=0$$ of the corresponding distribution. For a given sequence of real numbers $$(s_k)_{k\geq0}$$, the moment problem means to find the distribution $$\sigma$$. Depending on the supplementary properties of the sequence $$(s_k)_{k\geq0}$$, one may have a unique solution (modulo a constant at the points where it is continuous), several solutions or no solution at all. In modern terms, the distribution is replaced by a positive measure, called a representing measure, and the problem is said to be determinate, when the representing measure is uniquely determined.
A first extension of Stieltjes moment problem is due to H. Hamburger (1919–1921), who replaced the interval $$[0,\infty)$$ by the real line. Paradoxically, Hamburger’s condition is simpler than Stieltjes’ condition to get solutions of the corresponding moment problem. Other contributions, practically in the same period, are due to P. Nevanlinna, M. Riesz, E. Hellinger and T. Carleman, who found new methods of investigating the moment problems, and interesting connections with other branches of analysis (and not only) as well. All these mathematicians are the predecessors of many contributors to what is today known as the one-dimensional moment problem, whose treatment a consistent part of the monograph under review is dedicated.
Contributions concerning the moment problems in more than one variables, the so-called multidimensional moment problem, appear seemingly in the early thirties, related to the names of T. H. Hildebrandt and I. J. Schoenberg. Another consistent part of the book deals with this type of questions, presenting contributions by B. Fuglede, A. Devinatz, A. E.Nussbaum and others.
The monograph under review, published in the GTM collection of Springer, is much more than a book dedicated to graduate students. It contains useful results not only for young researchers but also for confirmed ones. It has an introductory part consisting of a preface and an overview, which presents some historical information, motivations of the study of moment problems, and connections with other chapters and domains of mathematics. There are two chapters, followed by four great parts, each of them dealing with a different branch of the theory of moment problems. An appendix and a comprehensive list of contributions are also included, and the book ends with a symbol index and an index of terms.
Because, roughly speaking, a moment problem means to find an integral representation for some objects as numerical sequences, more generally sequences of matrices or operators, or even for some special (spaces of) functions, Chapter 1 is dedicated to the integral representations of linear functionals via positive measures. The second chapter has a more abstract character, dealing with moment problems in abelian $$*$$-semigroups.
The one-dimensional moment problem is the subject of the first part of the book. Divided into six chapters, this part is a comprehensive presentation of various aspects of moment problems on finite or infinite intervals, including the real line. The role of positive polynomials on intervals, conditions insuring the determinacy, orthogonal polynomials and Jacobi operators are investigated. Then Hamburger and Stieltjes moment problems are treated via operator-theoretic approaches.
The second part, consisting of four chapters, is dedicated to the one-dimensional truncated moment problems, that is, roughly speaking, moment problems for finitely many data. After a general approach to this type of problems, the author presents two important particular cases, concerning the bounded intervals and the unit circle.
The third part, divided into five chapters, deals with the multidimensional moment problems, that is, moment problems depending on several variables. After a presentation of such problems on compact and on semi-algebraic sets, directions in which the author himself has important contributions, with results implying the existence or determinacy, the author presents some important applications related to the semidefinite programming and polynomial optimizations.
The last part is dedicated to multidimensional truncated moment problems, which is still a very active domain. One should mention one of the contributions of R. E. Curto and L. Fialkow, the so-called concept of flat extension, which has been the starting point of the research of other specialists.
To help the readers to go deeper into the matter, the author has added, after every chapter, a list of exercises, more or less difficult. Certain historical notes, concerning the contributions previously discussed, also accompany each chapter. Finally, the monograph also contains an appendix, recalling, for the sake of the convenience of the reader, concepts and results of analysis, frequently used in the monograph. The bibliography includes many older and newer titles, the author being one of the main contributors in the last forty years.
The book is well-written and it can be considered as an up-to day reference book for the use of both beginners and confirmed researchers, oriented not only to the domain of moment problems but also to other branches of analysis, or of mathematics, in general.

##### MSC:
 44A60 Moment problems 44-02 Research exposition (monographs, survey articles) pertaining to integral transforms 14P10 Semialgebraic sets and related spaces 47A57 Linear operator methods in interpolation, moment and extension problems 44-03 History of integral transforms
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