Any multi-index sequence has an interpolating measure

TL;DR

This paper extends Boas's result showing that any multi-index sequence of real numbers can be represented by an integral with respect to a measure, generalizing the single-index case.

Contribution

It proves that multi-index sequences of real numbers can be represented by measures, extending Boas's single-index sequence result to higher dimensions.

Findings

Any multi-index sequence has an associated representing measure.

Finite multi-index sequences also admit such measures.

The result generalizes classical moment sequence representations.

Abstract

R. P. Boas showed that any single-index sequence $\left\{ \beta_i \right\}_{i=0}^\infty$ of real numbers can be represented as $\beta_i =\int_0^\infty x^i \, d\mu$ ($i=0,1,2,\ldots$), where $\mu$ is a signed measure. As Boas said his observation seemed to be quite unexpected; however, it is even possible to extend the result to any multi-index sequence of real numbers. In addition, we can also prove that any multi-index finite sequence admits a measure of a similar type.