# 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 ${\sum}_{k=0}^{\mathrm{\infty}}{a}_{k}$, ${\sum}_{k=0}^{\mathrm{\infty}}{b}_{k}$ is given by:

$$\sum _{k=0}^{\mathrm{\infty}}{c}_{k}=\left(\sum _{k=0}^{\mathrm{\infty}}{a}_{k}\right)\cdot \left(\sum _{k=0}^{\mathrm{\infty}}{b}_{k}\right)=\sum _{k=0}^{\mathrm{\infty}}\sum _{l=0}^{k}{a}_{l}{b}_{k-l}.$$ | (2) |

A sufficient condition for the resulting series ${\sum}_{k=0}^{\mathrm{\infty}}{c}_{k}$ to be absolutely convergent is that ${\sum}_{k=0}^{\mathrm{\infty}}{a}_{k}$ and ${\sum}_{k=0}^{\mathrm{\infty}}{b}_{k}$ both converge absolutely .

Title | Cauchy product |
---|---|

Canonical name | CauchyProduct |

Date of creation | 2013-03-22 13:37:14 |

Last modified on | 2013-03-22 13:37:14 |

Owner | msihl (2134) |

Last modified by | msihl (2134) |

Numerical id | 7 |

Author | msihl (2134) |

Entry type | Definition |

Classification | msc 40-00 |