Complex Moment Problem Research Articles

The Complex Moment Problem and Subnormality: A Polar Decomposition Approach

It has been known that positive definiteness does not guarantee a bisequence to be a complex moment. However, it turns out that positive definite extendibility does (Theorems 1 and 22), and this is the main theme of this paper. The main tool is, generally understood, polar decomposition. To strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context (Theorems 9 and 29). All this enables us to prove characterizations of subnormality of unbounded operators having invariant domain (Theorems 37 and 39) and their further applications (Theorems 41 and 43) and a description of the complex moment problem on real algebraic curves (Theorems 52 and 56). The latter question is completed in the Appendix, in which we relate the complex moment problem to the two-dimensional real one, with emphasis on real algebraic sets.

Solution of the truncated complex moment problem for flat data

Introduction Moment matrices Positive moment matrices and representing measures Existence of representing measures Extension of flat positive moment matrices Applications Generalizations to several variables References List of symbols.

CENTER POINTS, EQUILIBRIUM POSITIONS, AND THE OBNOXIOUS LOCATION PROBLEM

ABSTRACT. Real variable analysis has een used to great benefit in a variety of classical problems in location theory. In this paper we explore basic complex variable techniques in one formulation of the obnoxious location problem. A general definition of center points is first given and used to formulate several alternate versions of the obnoxious location problem. A logarithmic transformation is then used to demonstrate some equivalences between these families of distinct location problems (defined via center points). A prototype logarithmic potential function which results from this formulation is then investigated, and it is demonstrated that the extremal solutions with this objective reside on the boundary of its domain of definition. An application using zero‐ and one‐dimensional centers is discussed, and a generalization to the spatial obnoxious problem is also briefly examined. We define a zero‐dimensional center as a critical point of the logarithmic potential function, and it is shown that these centers are equivalent to the solutions of the Complex Moment Problem.

Indefinite sequences with a finite order singularity in the complex moment problem

In this paper, we consider indefinite sequences having singularity at the point ( = (1 . . . . . 1) ~ (I~ p of order at most 2x. We show that they are related to both the classical moment problem and the L6vy-Khinchin formula for negative definite sequences. We establish an integral representation for these sequences and characterize the exponentially bounded ones by means of finitely many positive definiteness conditions. Even in the case x = 1, our approach provides new L6vy-Khinchin type integral representations. Let No be the set of all non-negative integers and fix the positive integer p. We denote by ~ l -z , i ] the algebra of polynomials in z and ~ where z = (zl . . . . . zp ) e r p. This algebra acts on the sequences ct = ~(m, n), (m,n)e ]Ng • N~ following the correspondence

Locating an obnoxious facility

We demonstrate that a formulation of the obnoxious single facility location problem is equivalent to the complex moment problem. The equivalence is understood in the sense that the critical points of the potential function defining the location problem are equal to the zeros of the function defining the complex moment problem.

The two-sided complex moment problem

The two-sided complex moment problem