##
**The problem of moments. Reprinted.**
*(English)*
Zbl 0041.43302

Mathematical Surveys, No. 1. New York: American Mathematical Society (AMS). xiv, 144 p. (1950).

This monograph is the first systematic account of its subject. To be able to treat the great material in a space of 134 pages, some special topics, such as the connections with Jacobi matrices and operators in Hilbert space, had to be left out and a few results are stated without proof.

An introductory chapter gives a short historical account of the moment problem, some facts on distribution functions, especially the Helly theorems are stated and a fundamental result on the extension of positive linear functionals is proved.

The first chapter applies the extension theorem and Helly’s theorem to prove that the moment problem \[ \int_{\mathbb R} u^i v^j \,d\Phi(u,v) = \mu_{ij}\quad (i,j = 0, 1, 2, \ldots) \] has a solution if and only if for every polynomial \(\sum a_{ij}u^iv^j\) which is \(\ge 0\) on \(\mathbb R\) we have \(\sum a_{ij} \mu_{ij}\ge 0\). The specialization of this result yields the classical criteria concerning the possibility of Hamburger’s, Stieltjes’, Hausdorff’s and the trigonometric moment problem. Next the uniqueness of the solution is investigated, a rather inelegant proof of Carleman’s sufficient conditions \((\sum \mu_{2n}^{-1/2n} = \infty\) for the Hamburger case) is given and the chapter closes with examples of indetermined moment problems.

The second chapter exposes the theory of the Hamburger moment problem. lt contains the classical results of Hamburger, R. Nevanlinna and M. Riesz, especially those concerning the determinacy of the problem, and the treatment follows very closely that of M. Riesz.

The third chapter bears the title: “Various modifications of the moment problem” and the following topics are discussed:

1. The problem of Markoff concerning the estimate of \(\displaystyle \int_{-1}^x f(t)\,d\psi(t)\) when the moments \(\displaystyle \int_{-1}^1 t^\nu\,d\psi(t) = \mu_\nu\) \(\nu = 0, 1, 2, \ldots, n)\) are known.

2. The conduction due to Markoff, Achyeser (Akhiezer) and Krein that the Hamburger problem admits an absolutely continuous solution.

3. The theory of Hausdorff’s moment problem and its connexions with completely monotonic sequences and functions. Explicit formulae and constructions for the solutions are given and also the criteria that \(\displaystyle \int_0^1 t^n\varphi(t)\,dt = \mu_n\) admit a solution \(\varphi(t)\in L^p\) \((1 < p\le \infty)\) are proved.

4. Boas’ theorem that \(\displaystyle \int_{-\infty}^\infty t^n\,d\alpha(t) = \mu_n\) admits always infinitely many solutions \(\alpha(t)\) of bounded variation.

5. Results are stated concerning problems in which \(t^n\) is replaced by \(t^{k_n}\).

The last chapter deals with approximate quadratures. Let \(q(t)\) be a quasi-orthogonal polynomial of degree \(n+1\) and of order \(n+1\) belonging to the moment problem \(\displaystyle \int_{-\infty}^\infty t^n\,d\psi(t) = \mu_n\).

Let \(x_0, x_1, \ldots, x_n\) be the roots of \(q(t)\), put \(q_j(t) = q(t)/(t - x_j) q'(x_j)\) \((j = 0, 1,\ldots, n)\). Let \(\psi_n(t)\) satisfy the “reduced moment problem” \(\displaystyle \int_{-\infty}^\infty t^\nu\,d\psi_n(t) = \mu_\nu\) \((\nu = 0, 1, \ldots, n)\) and let \(f(t)\) be any function defined in \(-\infty< t < \infty\). Then the convergence of \[ \int_{-\infty}^\infty \sum_{j=0}^n q_j(t)f(x_j) \,d\psi_n(t) = \sum_{j=0}^n \rho_n(x_j)f(x_j) \] to the value \(\displaystyle \int_{-\infty}^\infty f(t)\,d\psi(t)\) is discussed.

In the new printing a few corrections have been made and a useful supplementary bibliography has been added by R. P. Boas.

An introductory chapter gives a short historical account of the moment problem, some facts on distribution functions, especially the Helly theorems are stated and a fundamental result on the extension of positive linear functionals is proved.

The first chapter applies the extension theorem and Helly’s theorem to prove that the moment problem \[ \int_{\mathbb R} u^i v^j \,d\Phi(u,v) = \mu_{ij}\quad (i,j = 0, 1, 2, \ldots) \] has a solution if and only if for every polynomial \(\sum a_{ij}u^iv^j\) which is \(\ge 0\) on \(\mathbb R\) we have \(\sum a_{ij} \mu_{ij}\ge 0\). The specialization of this result yields the classical criteria concerning the possibility of Hamburger’s, Stieltjes’, Hausdorff’s and the trigonometric moment problem. Next the uniqueness of the solution is investigated, a rather inelegant proof of Carleman’s sufficient conditions \((\sum \mu_{2n}^{-1/2n} = \infty\) for the Hamburger case) is given and the chapter closes with examples of indetermined moment problems.

The second chapter exposes the theory of the Hamburger moment problem. lt contains the classical results of Hamburger, R. Nevanlinna and M. Riesz, especially those concerning the determinacy of the problem, and the treatment follows very closely that of M. Riesz.

The third chapter bears the title: “Various modifications of the moment problem” and the following topics are discussed:

1. The problem of Markoff concerning the estimate of \(\displaystyle \int_{-1}^x f(t)\,d\psi(t)\) when the moments \(\displaystyle \int_{-1}^1 t^\nu\,d\psi(t) = \mu_\nu\) \(\nu = 0, 1, 2, \ldots, n)\) are known.

2. The conduction due to Markoff, Achyeser (Akhiezer) and Krein that the Hamburger problem admits an absolutely continuous solution.

3. The theory of Hausdorff’s moment problem and its connexions with completely monotonic sequences and functions. Explicit formulae and constructions for the solutions are given and also the criteria that \(\displaystyle \int_0^1 t^n\varphi(t)\,dt = \mu_n\) admit a solution \(\varphi(t)\in L^p\) \((1 < p\le \infty)\) are proved.

4. Boas’ theorem that \(\displaystyle \int_{-\infty}^\infty t^n\,d\alpha(t) = \mu_n\) admits always infinitely many solutions \(\alpha(t)\) of bounded variation.

5. Results are stated concerning problems in which \(t^n\) is replaced by \(t^{k_n}\).

The last chapter deals with approximate quadratures. Let \(q(t)\) be a quasi-orthogonal polynomial of degree \(n+1\) and of order \(n+1\) belonging to the moment problem \(\displaystyle \int_{-\infty}^\infty t^n\,d\psi(t) = \mu_n\).

Let \(x_0, x_1, \ldots, x_n\) be the roots of \(q(t)\), put \(q_j(t) = q(t)/(t - x_j) q'(x_j)\) \((j = 0, 1,\ldots, n)\). Let \(\psi_n(t)\) satisfy the “reduced moment problem” \(\displaystyle \int_{-\infty}^\infty t^\nu\,d\psi_n(t) = \mu_\nu\) \((\nu = 0, 1, \ldots, n)\) and let \(f(t)\) be any function defined in \(-\infty< t < \infty\). Then the convergence of \[ \int_{-\infty}^\infty \sum_{j=0}^n q_j(t)f(x_j) \,d\psi_n(t) = \sum_{j=0}^n \rho_n(x_j)f(x_j) \] to the value \(\displaystyle \int_{-\infty}^\infty f(t)\,d\psi(t)\) is discussed.

In the new printing a few corrections have been made and a useful supplementary bibliography has been added by R. P. Boas.

Reviewer: J. HorvĂˇth

### MSC:

44A60 | Moment problems |

44-02 | Research exposition (monographs, survey articles) pertaining to integral transforms |

42A70 | Trigonometric moment problems in one variable harmonic analysis |