Logarithmic convolution

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the scale convolution of two <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> 
  
    
      
        s
        (
        t
        )
      
    
    {\displaystyle s(t)}
  
 and 
  
    
      
        r
        (
        t
        )
      
    
    {\displaystyle r(t)}
  
, also known as their logarithmic convolution or log-volution is defined as the function
</p>

s
        
          ∗
          
            l
          
        
        r
        (
        t
        )
        =
        r
        
          ∗
          
            l
          
        
        s
        (
        t
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        s
        
          (
          
            
              t
              a
            
          
          )
        
        r
        (
        a
        )
        
        
          
            
              d
              a
            
            a
          
        
      
    
    {\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}}

<p>when this quantity exists.
</p>

Logarithmic convolution open-in-new

Logarithmic convolution