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Homelogarithmic convolution

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# 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

$s\ast_{l}r(t)=r\ast_{l}s(t)=\int_{0}^{\infty}s(\frac{t}{a})r(a)\frac{da}{a}$ |

when this quantity exists.

# Results

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

$\displaystyle s\ast_{l}r(t)$ | $\displaystyle=$ | $\displaystyle\int_{0}^{\infty}s(\frac{t}{a})r(a)\frac{da}{a}=\int_{{-\infty}}^% {\infty}s(\frac{t}{e^{u}})r(e^{u})du$ | ||

$\displaystyle=$ | $\displaystyle\int_{{-\infty}}^{\infty}s(e^{{\log t-u}})r(e^{u})du$ |

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

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

Related:

Convolution

Synonym:

scale convolution

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

44A35*no label found*

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