A note on Hausdorff's summation methods

Joseph Patrick Brannen

Vol. 15 (1965), No. 1, 29–33

DOI: 10.2140/pjm.1965.15.29

Abstract

If $\left\{a_n\right\}$ is a moment sequence and $\left(\Delta a\right)$ is the difference matrix having base sequence $\left\{a_n\right\}$, then $\left(\Delta a\right)$ is symmetric about the main diagonal if and only if the function $\alpha \left(x\right)$ such that $a_n={\int \; <!--nolimits-->}_0^1{x}^{n}d\alpha \left(x\right),n=0,1,2,\cdots$, is symmetric in the sense that $\alpha \left(x\right)+\alpha \left(1+x\right)=\alpha \left(1\right)+\alpha \left(0\right)$ except for at most countably many $x$ in $\left[0,1\right]$. This property is related to the “fixed points” of the matrix $H$, where $HaH$ is the Hausdorff matrix determined by the moment sequence $\left\{a_n\right\}$.

Mathematical Subject Classification

Primary: 40.30

Milestones

Received: 15 March 1964

Published: 1 March 1965