*KCI등재*

*학술저널*

#
*CONSISTENCY AND GENERAL TRUNCATED MOMENT PROBLEMS*

- 충청수학회
- Journal of the Chungcheong Mathematical Society
- Volume 31, No. 4
- 등재여부 : KCI등재
- 2018.11
- 487 - 509 (23 pages)

The Truncated Moment Problem (TMP) entails finding a positive Borel measure to represent all moments in a finite sequence as an integral; once the sequence admits one or more such measures, it is known that at least one of the measures must be finitely atomic with positive densities (equivalently, a linear combination of Dirac point masses with positive coefficients). On the contrary, there are more general moment problems for which we aim to find a signed measure to represent a sequence; that is, the measure may have some negative densities. This type of problem is referred to as the General Truncated Moment Problem (GTMP). The Jordan Decomposition Theorem states that any (signed) measure can be written as a difference of two positive measures, and hence, in the view of this theorem, we are able to apply results for TMP to study GTMP. In this note we observe differences between TMP and GTMP; for example, we cannot have an analogous to the Flat Extension Theorem for GTMP. We then present concrete solutions to lower-degree problems.

Abstract

1. Introduction

1.1. Definitions and Notations

1.2. Necessary Conditions for TMP

1.3. Flat Extension

2. The Consistency and a Measure

3. Vandermonde Matrix Method to Check the Consistency

4. General Quadratic Moment Problem

5. General Quartic Moment Problem

6. Conclusions and Remarks

References