Logarithmic convolution - Alchetron, the free social encyclopedia

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

s ∗ l r ( t ) = r ∗ l s ( t ) = ∫ 0 ∞ s ( t a ) r ( a ) d a 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 :

s ∗ l r ( t ) = ∫ 0 ∞ s ( t a ) r ( a ) d a a = ∫ − ∞ ∞ s ( t e u ) r ( e u ) d u = ∫ − ∞ ∞ s ( e log ⁡ t − u ) r ( e u ) d u .

Define f ( v ) = s ( e v ) and g ( v ) = r ( e v ) and let v = log ⁡ t , then

s ∗ l r ( v ) = f ∗ g ( v ) = g ∗ f ( v ) = r ∗ l s ( v ) .

This article incorporates material from logarithmic convolution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

References

Logarithmic convolution Wikipedia

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