Any multi-index sequence has an interpolating measure

Abstract

R. P. Boas showed that any single-index sequence ${\left\{{\beta }_i\right\}}_{i=0}^{\infty }$ of real numbers can be represented as ${\beta }_i={\int }_0^{\infty }{x}^{i}d\mu$ ($i=0,1,2,\dots$), 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.