40-00-CauchyProduct.tex

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\pmcreated{2013-03-22 13:37:14}
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\pmtitle{Cauchy product}
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\pmtype{Definition}
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\begin{document}
Let $a_k$ and $b_k$ be two sequences of real or complex numbers for 
$k \in {\mathbb N}_0$ ( ${\mathbb N}_0$ is the set of natural numbers containing zero).
The Cauchy product is defined by:
\begin{equation}
(a \circ b)(k) = \sum_{l=0}^k a_l b_{k-l}.
\end{equation}
This is basically the convolution for two sequences.
Therefore the product of two series $\sum_{k=0}^{\infty} a_k$, $\sum_{k=0}^{\infty} b_k$ is given by:
\begin{equation}
\sum_{k=0}^{\infty} c_k = \left(\sum_{k=0}^{\infty} a_k \right)\cdot \left(\sum_{k=0}^{\infty} b_k \right) =
\sum_{k=0}^{\infty} \sum_{l=0}^k a_l b_{k-l}.
\end{equation}
A sufficient condition for the resulting series $\sum_{k=0}^{\infty} c_k$ to be absolutely convergent is that $\sum_{k=0}^{\infty} a_k$ and $\sum_{k=0}^{\infty} b_k$ both converge absolutely .
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