# logarithmic convolution

## Definition

The *scale convolution* of two functions $s(t)$ and $r(t)$, also known as their *logarithmic convolution* is defined as the function

$$s{\ast}_{l}r(t)=r{\ast}_{l}s(t)={\int}_{0}^{\mathrm{\infty}}s(\frac{t}{a})r(a)\frac{da}{a}$$ |

when this quantity exists.

## Results

The logarithmic convolution can be related to the ordinary convolution by changing the variable from $t$ to $v=\mathrm{log}t$:

$s{\ast}_{l}r(t)$ | $=$ | ${\int}_{0}^{\mathrm{\infty}}}s({\displaystyle \frac{t}{a}})r(a){\displaystyle \frac{da}{a}}={\displaystyle {\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}}s({\displaystyle \frac{t}{{e}^{u}}})r({e}^{u})\mathit{d}u$ | ||

$=$ | ${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}}s({e}^{\mathrm{log}t-u})r({e}^{u})\mathit{d}u$ |

Define $f(v)=s({e}^{v})$ and $g(v)=r({e}^{v})$ and let $v=\mathrm{log}t$, then

$$s{\ast}_{l}r(v)=f\ast g(v)=g\ast f(v)=r{\ast}_{l}s(v).$$ |

Title | logarithmic convolution |
---|---|

Canonical name | LogarithmicConvolution |

Date of creation | 2013-03-22 14:28:26 |

Last modified on | 2013-03-22 14:28:26 |

Owner | swiftset (1337) |

Last modified by | swiftset (1337) |

Numerical id | 4 |

Author | swiftset (1337) |

Entry type | Definition |

Classification | msc 44A35 |

Synonym | scale convolution |

Related topic | Convolution |