Convolution power

<h2 id="properties">Properties</h2>
If x is itself suitably differentiable, then from the <a href="/facts/Convolution/PVPrdz9J">properties</a> of convolution, one has

D
          
        
        
          
            {
          
        
        
          x
          
            ∗
            n
          
        
        
          
            }
          
        
        =
        (
        
          
            D
          
        
        x
        )
        ∗
        
          x
          
            ∗
            (
            n
            −
            1
            )
          
        
        =
        x
        ∗
        
          
            D
          
        
        
          
            {
          
        
        
          x
          
            ∗
            (
            n
            −
            1
            )
          
        
        
          
            }
          
        
      
    
    {\displaystyle {\mathcal {D}}{\big \{}x^{*n}{\big \}}=({\mathcal {D}}x)*x^{*(n-1)}=x*{\mathcal {D}}{\big \{}x^{*(n-1)}{\big \}}}

where 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
 denotes the <a href="/facts/Derivative/7Ito70Dh">derivative</a> operator. Specifically, this holds if x is a compactly supported distribution or lies in the <a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> W1,1 to ensure that the derivative is sufficiently regular for the convolution to be well-defined.

<h2 id="applications">Applications</h2>
In the configuration random graph, the size distribution of <a href="/facts/Connected_component_(graph_theory)/5owVSXtY">connected components</a> can be expressed via the convolution power of the excess <a href="/facts/Degree_distribution/qVlXFfbA">degree distribution</a> (Kryven (2017)):

w
        (
        n
        )
        =
        
          
            {
            
              
                
                  
                    
                      
                        μ
                        
                          1
                        
                      
                      
                        n
                        −
                        1
                      
                    
                  
                  
                    u
                    
                      1
                    
                    
                      ∗
                      n
                    
                  
                  (
                  n
                  −
                  2
                  )
                  ,
                
                
                  n
                  >
                  1
                  ,
                
              
              
                
                  u
                  (
                  0
                  )
                
                
                  n
                  =
                  1.
                
              
            
            
          
        
      
    
    {\displaystyle w(n)={\begin{cases}{\frac {\mu _{1}}{n-1}}u_{1}^{*n}(n-2),&n>1,\\u(0)&n=1.\end{cases}}}

Here, 
 
 
 
 w
 (
 n
 )
 
 
 {\displaystyle w(n)}
 
 is the size distribution for connected components, 
 
 
 
 
 u
 
 1
 
 
 (
 k
 )
 =
 
 
 
 k
 +
 1
 
 
 μ
 
 1
 
 
 
 
 u
 (
 k
 +
 1
 )
 ,
 
 
 {\displaystyle u_{1}(k)={\frac {k+1}{\mu _{1}}}u(k+1),}
 
 is the excess degree distribution, and 
 
 
 
 u
 (
 k
 )
 
 
 {\displaystyle u(k)}
 
 denotes the <a href="/facts/Degree_distribution/qVlXFfbA">degree distribution</a>.
As <a href="/facts/Convolution_algebra/I1BVHqYD">convolution algebras</a> are special cases of <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebras</a>, the convolution power is a special case of the (ordinary) power in a Hopf algebra. In applications to <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, the convolution exponential, convolution logarithm, and other analytic functions based on the convolution are constructed as formal power series in the elements of the algebra (Brouder, Frabetti & Patras 2008). If, in addition, the algebra is a <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a>, then convergence of the series can be determined as above. In the formal setting, familiar identities such as

x
        =
        
          log
          
            ∗
          
        
        ⁡
        (
        
          exp
          
            ∗
          
        
        ⁡
        x
        )
        =
        
          exp
          
            ∗
          
        
        ⁡
        (
        
          log
          
            ∗
          
        
        ⁡
        x
        )
      
    
    {\displaystyle x=\log ^{*}(\exp ^{*}x)=\exp ^{*}(\log ^{*}x)}

continue to hold. Moreover, by the permanence of functional relations, they hold at the level of functions, provided all expressions are well-defined in an open set by convergent series.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Convolution/PVPrdz9J">Convolution</a></li>
<li><a href="/facts/Convolution_theorem/E7zhjIx2">Convolution theorem</a></li>
<li><a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a></li>
<li><a href="/facts/Taylor_series/M0aTuftH">Taylor series</a></li></ul>

<ul><li>Schwartz, Laurent (1951), Théorie des Distributions, Tome II, Herman, Paris.</li>
<li><a href="/facts/John_Horvath_(mathematician)/VcacePH8">Horváth, John</a> (1966), Topological Vector Spaces and Distributions, Addison-Wesley Publishing Company: Reading, MA, USA.</li>
<li>Ben Chrouda, Mohamed; El Oued, Mohamed; Ouerdiane, Habib (2002), "Convolution calculus and applications to stochastic differential equations", Soochow Journal of Mathematics, 28 (4): 375–388, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0250-3255">0250-3255</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1953702">1953702</a>.</li>
<li>Brouder, Christian; Frabetti, Alessandra; Patras, Frédéric (2008). "Decomposition into one-particle irreducible Green functions in many-body physics". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0803.3747">0803.3747</a> [<a href="https://arxiv.org/archive/cond-mat.str-el">cond-mat.str-el</a>]..</li>
<li><a href="/facts/William_Feller/CeLs6QpB">Feller, William</a> (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0270403">0270403</a>.</li>
<li>Stroock, Daniel W. (1993), Probability theory, an analytic view, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-43123-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1267569">1267569</a>.</li>
<li>Kryven, I (2017), "General expression for component-size distribution in infinite configuration networks", Physical Review E, 95 (5): 052303, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1703.05413">1703.05413</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2017PhRvE..95e2303K">2017PhRvE..95e2303K</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2Fphysreve.95.052303">10.1103/physreve.95.052303</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/28618550">28618550</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:8421307">8421307</a>.</li></ul>

Convolution power open-in-new

Convolution power