The moment problem

The moment problem, or one should say moment problems (plural) because there are several different classical moment problems. Some ideas can be found in work of Chebyshev and Markov, but Stieltjes at the end of the nineteenth century was one of the first to formally consider the moment problem named after him. Given a sequence of numbers ($m_k$), is there a positive measure $\mu$ such that $m_k=\int x^k \mu(dx), k=0,1,2,\ldots$? In the case of Stieltjes, the measure was supposed to have a support on the positive real line. First of all one wants to find out under what conditions such a measure exists, then when the solution is unique, and when it is not unique to characterize all possible solutions. Soon (around 1920) other versions were formulated by Hamburger (when the support of the measure is the whole real line) and Hausdorff (when the support is a finite interval) and some ten years later the trigonometric moment problem was tackled by Verblunsky, Akhiezer and Krein where the support is the complex unit circle. There is a basic difference between the trigonometric moment problem and the other classical moment problems on (parts of) the real line. In the latter situation, the existence of a solution is guaranteed by requiring the positivity of Hankel matrices whose entries are the moments. In the trigonometric case, the Hankel matrices are replaced by Toeplitz matrices. The latter involve also the moments $m_{-k}=\overline{m}_k, k=1,2,\ldots$ which are automatically matched as well. Not so for the other moment problems. When also imposing moments with a negative index in those cases, this is called a strong moment problem. When only a finite number of moments are prescribed, this is called a truncated moment problem.

The importance of the moment problem is a consequence of the fact that it is at the crossroads of several branches and applications of mathematics. It relates to linear algebra, functional analysis and operator theory, stochastic processes, approximation theory, optimization, orthogonal polynomials, systems theory, scattering theory, signal processing, probability, and many more. No wonder that the greatest names in mathematics have contributed to the problem with papers and monographs. Because of the many connections to different fields also many approaches and many generalizations have been considered. The previously described moments are called power moments because of the $x^k$, but one could also prescribe moments based on a set of other functions $M_k(x)$. Traditionally, the Hausdorff moment problem is formulated for the interval [0,1], but one may consider any finite interval $[a,b]$ just like the Stieltjes moment problem could be formulated for any half line $[\alpha,\infty)$. Other generalizations lifts these problems to a block version, by assuming that the moments are matrices and the measure is matrix-valued, or the variable $x$ can have several components, resulting in a multivariate moment problem.

The fact that today, 100 years after Hamburger and Hausdorff, this is still an active research field is another proof of the importance of moment problems. Many books did appear already that were devoted to moment problems or where moment problems played an essential role. Some classics are Shohat and Tamarkin The Problem of Moments (1943), Akhiezer The classical moment problem and some related questions in analysis (1965), Krein and Nudelman The Markov moment problem and extremal problems (1977). The present book is a modern update of the situation. It gives an operator theoretic approach to moment problems, leaving aside the applications. The univariate classical problems of Hamburger ($\mathbb{R}$), Stieltjes ($[0,\infty)$) and Hausdorff ($[a,b]$), appear both in their full and their truncated version. Also the trigonometric moment problem is represented but by only one chapter.
The introduction to these problems is quite general. It is showing how integral representations for linear functionals can be obtained, and in particular how this works for finite dimensional spaces, and for truncated moment problems. Another essential tool is giving some examples of how moment problems can be defined on a commutative *-semigroup. Indeed, all what is needed is a structure with an involution (which could be the identity) and it should allow the definition of a positive definite linear functional so that it can give rise to an inner product on the space of polynomials (and its completion). With gross oversimplification one could say that a sequence is a moment sequence if the associated linear functional is positive and the solution corresponds to the measure that appears in an integral representation of the functional. For real problems, the involution is the identity: $x^*=x$, for complex problems, the involution $x^*=1/\overline{x}$ allows to treat the trigonometric moment problem at the same level as the real moment problems.
This general approach is not really needed for the classical one dimensional moment problems that are treated in part I and the truncated version in part II, but the generality of the introduction allows more easy generalizations to the multivariate case and its truncated version that are discussed in parts III and IV respectively. What is treated in the first two parts are the classical results: the representation of positive polynomials, conditions for the existence of a solution of the moment problem, Hankel matrices, orthogonal polynomials and the Jacobi operator, determinacy (i.e. uniqueness) of the solution, the characterization of all solutions in the indeterminate case, and the relation with complex interpolation problems for Pick functions. For truncated moment problems one may look for some special, so called N-extremal, solutions which lie on the boundary of the solution set, or a canonical solution or solutions that maximize the mass in a particular point of an atomic solution.

For the multivariate case, it takes some more work and we do not have the classical cases where the measure should be supported and generalizations can go in many different directions. Nevertheless, the corresponding chapters in parts III and IV go through the same steps as in the univariate case as much as possible. What are representing measures and when are polynomials positive? By defining the moment problem for a finitely generated abelian unital algebra, and using a fiber theorem that characterizes moment functionals, some generalizations of the one-dimensional case can be obtained (like for example a rational moment problem) or moment problems on some cubics. Determinacy of the multivariate moment problem is given in the form of a generalized Carleman condition, moments for the Gaussian measure on the unit sphere, and complex one- and two-sided moment problems are all discussed. Characterizing a canonical or extreme solution(s) is not as simple as in the one-dimensional case. Only for the truncated multivariate problems Hankel matrices are introduced and atomic solutions with maximization of a point mass can be characterized.

The book appears in the series Graduate Texts in Mathematics which means that it is conceived as a as a text that could be used for lecturing with proofs fully included and extra exercises after every chapter as well as notes the refer to the history and the related literature. It is however marvellously capturing the present state of the art of the topic. So it will be also a reference work for researchers. It captures a survey of the univariate case and indicates research directions for the multivariate problem. The list of references at the end of the book has both historical as recent publications, but it is restricted to what has been discussed in the present book. Schmüdgen has published two books before on operator theory, so he knows how to write a book on a difficult subject and still keep it accessible for the audience that he is addressing (graduate students and researchers). Lists of symbols are really helpful to remember notation. The fact that on page 4 Chebyshev and Markov are situated in 1974 and 1984 respectively is just a glitch in an otherwise carefully edited text.

Adhemar Bultheel
Book details

This is a modern operator approach surveying classical one-dimensional moment problems, but the setting is general by formulating the problem on an abelian *-semigroups. This allows to also capture an introduction to multivariate moment problems which is much more recent and a subject that is still in evolution. The characterization of moment sequences, associated linear moment functionals, and determinate as well as indeterminate problems for the full or the truncated problems are discussed. Particular canonical and N-extremal solutions or solutions with a maximal mass point are discussed.

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978-3-319-64545-2 (hbk); 978-3-319-64546-9 (ebk)
84,79 € (hbk); 67,82 € (ebk)