Poisson summation formula

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Poisson summation formula is an equation that relates the <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> coefficients of the <a href="/facts/Periodic_summation/x3jMCU4y">periodic summation</a> of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> to values of the function's <a href="/facts/Continuous_Fourier_transform/XXhKJtc8">continuous Fourier transform</a>. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by <a href="/facts/Sim%25C3%25A9on_Denis_Poisson/zmsDITrY">Siméon Denis Poisson</a> and is sometimes called Poisson resummation.
</p><p>For a smooth, complex valued function 
  
    
      
        s
        (
        x
        )
      
    
    {\displaystyle s(x)}
  
 on 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 which decays at infinity with all derivatives (<a href="/facts/Schwartz_function/B035faw7">Schwartz function</a>), the simplest version of the Poisson summation formula states that
</p>

<table><tbody><tr><td>                              ∑                      n            =            −            ∞                                ∞                          s        (        n        )        =                  ∑                      k            =            −            ∞                                ∞                          S        (        k        )        .              {\displaystyle \sum _{n=-\infty }^{\infty }s(n)=\sum _{k=-\infty }^{\infty }S(k).}      </td> <td></td> <td>Eq.1</td></tr></tbody></table>

<p>where 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of 
  
    
      
        s
      
    
    {\displaystyle s}
  
, i.e., 
  
    
      
        S
        (
        f
        )
        ≜
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        s
        (
        x
        )
         
        
          e
          
            −
            i
            2
            π
            f
            x
          
        
        
        d
        x
        .
      
    
    {\textstyle S(f)\triangleq \int _{-\infty }^{\infty }s(x)\ e^{-i2\pi fx}\,dx.}
  
 The summation formula can be restated in many equivalent ways, but a simple one is the following. Suppose that 
  
    
      
        f
        ∈
        
          L
          
            1
          
        
        (
        
          
            R
          
          
            n
          
        
        )
      
    
    {\displaystyle f\in L^{1}(\mathbb {R} ^{n})}
  
 (<i>L</i>1 for <a href="/facts/Lp_space/gbad0Rv1"><i>L</i>1 space</a>) and 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
 is a <a href="/facts/Unimodular_lattice/6r8l4y81">unimodular lattice</a> in 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
. Then the periodization of 
  
    
      
        f
      
    
    {\displaystyle f}
  
, which is defined as the sum 
  
    
      
        
          f
          
            Λ
          
        
        (
        x
        )
        =
        
          ∑
          
            λ
            ∈
            Λ
          
        
        f
        (
        x
        +
        λ
        )
        ,
      
    
    {\textstyle f_{\Lambda }(x)=\sum _{\lambda \in \Lambda }f(x+\lambda ),}
  
 converges in the 
  
    
      
        
          L
          
            1
          
        
      
    
    {\displaystyle L^{1}}
  
 norm of 
  
    
      
        
          
            R
          
          
            n
          
        
        
          /
        
        Λ
      
    
    {\displaystyle \mathbb {R} ^{n}/\Lambda }
  
 to an 
  
    
      
        
          L
          
            1
          
        
        (
        
          
            R
          
          
            n
          
        
        
          /
        
        Λ
        )
      
    
    {\displaystyle L^{1}(\mathbb {R} ^{n}/\Lambda )}
  
 function having Fourier series 
  
    
      
        
          f
          
            Λ
          
        
        (
        x
        )
        ∼
        
          ∑
          
            
              λ
              ′
            
            ∈
            
              Λ
              ′
            
          
        
        
          
            
              f
              ^
            
          
        
        (
        
          λ
          ′
        
        )
        
          e
          
            2
            π
            i
            
              λ
              ′
            
            x
          
        
      
    
    {\displaystyle f_{\Lambda }(x)\sim \sum _{\lambda '\in \Lambda '}{\hat {f}}(\lambda ')e^{2\pi i\lambda 'x}}
  
 where 
  
    
      
        
          Λ
          ′
        
      
    
    {\displaystyle \Lambda '}
  
 is the dual lattice to 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
. (Note that the Fourier series on the right-hand side need not converge in 
  
    
      
        
          L
          
            1
          
        
      
    
    {\displaystyle L^{1}}
  
 or otherwise.)
</p>

Poisson summation formula open-in-new

Poisson summation formula