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