Cauchy product

Let ak and bk be two sequencesPlanetmathPlanetmath of real or complex numbers for k0 ( 0 is the set of natural numbers containing zero). The Cauchy productMathworldPlanetmath is defined by:

(ab)(k)=l=0kalbk-l. (1)

This is basically the convolution for two sequences. Therefore the product of two series k=0ak, k=0bk is given by:

k=0ck=(k=0ak)(k=0bk)=k=0l=0kalbk-l. (2)

A sufficient condition for the resulting series k=0ck to be absolutely convergent is that k=0ak and k=0bk 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