Cauchy product

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:

 $(a\circ b)(k)=\sum_{l=0}^{k}a_{l}b_{k-l}.$ (1)

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:

 $\sum_{k=0}^{\infty}c_{k}=\left(\sum_{k=0}^{\infty}a_{k}\right)\cdot\left(\sum_% {k=0}^{\infty}b_{k}\right)=\sum_{k=0}^{\infty}\sum_{l=0}^{k}a_{l}b_{k-l}.$ (2)

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 .

Title Cauchy product CauchyProduct 2013-03-22 13:37:14 2013-03-22 13:37:14 msihl (2134) msihl (2134) 7 msihl (2134) Definition msc 40-00