Vol. 15, No. 1, 1965

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A note on Hausdorff's summation methods

Joseph Patrick Brannen

Vol. 15 (1965), No. 1, 29–33
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 ${a}_{n}={\int }_{0}^{1}\phantom{\rule{0.3em}{0ex}}{x}^{n}\phantom{\rule{0.3em}{0ex}}d\alpha \left(x\right),\phantom{\rule{1em}{0ex}}n=0,1,2,\cdots \phantom{\rule{0.3em}{0ex}}$, 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 $\left\{{a}_{n}\right\}$.

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