Extremal solutions of the two-dimensional \(L\)-problem of moments. II.

*(English)*Zbl 0910.47011This article is the continuation of the first article with the same title of the same author [published in J. Funct. Anal. 136, No. 2, 331-364 (1996)]. The purpose of this article is to apply the techniques developed in the first article to all extremal solutions of the \(L\)-problem.

We give the following definitions to introduce some of the important results. Let \(K\) be a compact subset of the complex plane, let \(L\) be a fixed positive constant, and let \(N\) be a fixed positive integer. We are interested in classifying and characterizing in intrinsic terms the moments \(a_{nm}= \int_K\varphi(x,y) x^ny^mdA\), \(0\leq m\leq m+ n\leq N\), of a measurable function \(\varphi\) on \(K\) which satisfies \(0\leq\varphi\leq L\), \(dA\)-a.e., where \(dA\) is the planar Lebesgue measure. We denote \(\sum\) the collection of all vectors \(a= (a_{kl})_{k+l\leq N}\in\mathbb{R}^d\), \(d={(N+1)(N+2)\over 2}\) which arise as the moments of a function \(\varphi\). The author follows methods of M. G. Krein’s convexity theory.

In this paper, Section 1 is an introduction. Section 2 reminds the main results of M. G. Krein which characterize the extremal solutions of the truncated \(L\)-problem of moments. In Section 3, the research of the associated hyponormal operators is presented and will be established via a rational cyclicity criterion, the existence of a general analytic model of them, valid for all extremal solutions described in Section 2. In Section 4, the canonical exponential kernel of an extremal solution of the momentum problem is analyzed.

The author presents an intrinsic characterization of all kernels and interprets the \(L\)-problem of moments as an interpolation problem of the Carathéodory-Fejér type, for a class of analytic functions defined in the bidisk.

We give the following definitions to introduce some of the important results. Let \(K\) be a compact subset of the complex plane, let \(L\) be a fixed positive constant, and let \(N\) be a fixed positive integer. We are interested in classifying and characterizing in intrinsic terms the moments \(a_{nm}= \int_K\varphi(x,y) x^ny^mdA\), \(0\leq m\leq m+ n\leq N\), of a measurable function \(\varphi\) on \(K\) which satisfies \(0\leq\varphi\leq L\), \(dA\)-a.e., where \(dA\) is the planar Lebesgue measure. We denote \(\sum\) the collection of all vectors \(a= (a_{kl})_{k+l\leq N}\in\mathbb{R}^d\), \(d={(N+1)(N+2)\over 2}\) which arise as the moments of a function \(\varphi\). The author follows methods of M. G. Krein’s convexity theory.

In this paper, Section 1 is an introduction. Section 2 reminds the main results of M. G. Krein which characterize the extremal solutions of the truncated \(L\)-problem of moments. In Section 3, the research of the associated hyponormal operators is presented and will be established via a rational cyclicity criterion, the existence of a general analytic model of them, valid for all extremal solutions described in Section 2. In Section 4, the canonical exponential kernel of an extremal solution of the momentum problem is analyzed.

The author presents an intrinsic characterization of all kernels and interprets the \(L\)-problem of moments as an interpolation problem of the Carathéodory-Fejér type, for a class of analytic functions defined in the bidisk.

Reviewer: A.Kondrat’yev (Pensacola)

##### MSC:

47A57 | Linear operator methods in interpolation, moment and extension problems |

##### Keywords:

moments; M. G. Krein’s convexity theory; truncated \(L\)-problem of moments; analytic model; exponential kernel of an extremal solution of the momentum problem; interpolation problem of the Carathéodory-Fejér type
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