Recursively determined representing measures for bivariate truncated moment sequences

TL;DR

This paper investigates conditions under which a bivariate truncated moment sequence admits a representing measure, focusing on recursively generated extensions of the associated moment matrix and providing constructive procedures for verification.

Contribution

It establishes that for bivariate recursively determinate moment matrices, the existence of certain positive extensions guarantees a representing measure, and details a constructive method to verify these extensions.

Findings

Existence of positive, recursively generated extensions implies the existence of a measure.

All extensions up to a certain order may be necessary to confirm a measure.

Constructive procedures can determine the existence of such extensions.

Abstract

A theorem of Bayer and Teichmann implies that if a finite real multisequence \beta = \beta^(2d) has a representing measure, then the associated moment matrix M_d admits positive, recursively generated moment matrix extensions M_(d+1), M_(d+2),... For a bivariate recursively determinate M_d, we show that the existence of positive, recursively generated extensions M_(d+1),...,M_(2d-1) is sufficient for a measure. Examples illustrate that all of these extensions may be required to show that \beta has a measure. We describe in detail a constructive procedure for determining whether such extensions exist. Under mild additional hypotheses, we show that M_d admits an extension M_(d+1) which has many of the properties of a positive, recursively generated extension.