Bump function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a bump function (also called a test function) is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
        :
        
          
            R
          
          
            n
          
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
  
 on a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 which is both <a href="/facts/Smooth_function/mffL4ch2">smooth</a> (in the sense of having <a href="/facts/Continuous_function/sKbl02pB">continuous</a> <a href="/facts/Derivative/7Ito70Dh">derivatives</a> of all orders) and <a href="/facts/Support_(mathematics)/BTuMGbsb">compactly supported</a>.  The <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of all bump functions with <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 forms a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>, denoted 
  
    
      
        
          
            C
          
          
            0
          
          
            ∞
          
        
        (
        
          
            R
          
          
            n
          
        
        )
      
    
    {\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}
  
 or 
  
    
      
        
          
            C
          
          
            
              c
            
          
          
            ∞
          
        
        (
        
          
            R
          
          
            n
          
        
        )
        .
      
    
    {\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}
  
  The <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of this space endowed with a suitable <a href="/facts/Topological_space/v2ut7CbK">topology</a> is the space of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a>.
</p>

Bump function open-in-new

Bump function