Weierstrass transform

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Weierstrass transform of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 f
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle f:\mathbb {R} \to \mathbb {R} }
 
, named after <a href="/facts/Karl_Weierstrass/4U8vdQzL">Karl Weierstrass</a>, is a "smoothed" version of 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 obtained by averaging the values of 
 
 
 
 f
 
 
 {\displaystyle f}
 
, weighted with a Gaussian centered at 
 
 
 
 x
 
 
 {\displaystyle x}
 
.

Specifically, it is the function 
 
 
 
 F
 
 
 {\displaystyle F}
 
 defined by

F
        (
        x
        )
        =
        
          
            1
            
              4
              π
            
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        f
        (
        y
        )
        
        
          e
          
            −
            
              
                
                  (
                  x
                  −
                  y
                  
                    )
                    
                      2
                    
                  
                
                4
              
            
          
        
        
        d
        y
        =
        
          
            1
            
              4
              π
            
          
        
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        f
        (
        x
        −
        y
        )
        
        
          e
          
            −
            
              
                
                  y
                  
                    2
                  
                
                4
              
            
          
        
        
        d
        y
         
        ,
      
    
    {\displaystyle F(x)={\frac {1}{\sqrt {4\pi }}}\int _{-\infty }^{\infty }f(y)\;e^{-{\frac {(x-y)^{2}}{4}}}\;dy={\frac {1}{\sqrt {4\pi }}}\int _{-\infty }^{\infty }f(x-y)\;e^{-{\frac {y^{2}}{4}}}\;dy~,}

the <a href="/facts/Convolution/PVPrdz9J">convolution</a> of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with the <a href="/facts/Gaussian_function/5bG8Kpp9">Gaussian function</a>

1
            
              4
              π
            
          
        
        
          e
          
            −
            
              x
              
                2
              
            
            
              /
            
            4
          
        
         
        .
      
    
    {\displaystyle {\frac {1}{\sqrt {4\pi }}}e^{-x^{2}/4}~.}

The factor 
 
 
 
 
 
 1
 
 4
 π
 
 
 
 
 
 {\displaystyle {\frac {1}{\sqrt {4\pi }}}}
 
 is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform.
Instead of 
 
 
 
 F
 (
 x
 )
 
 
 {\displaystyle F(x)}
 
 one also writes 
 
 
 
 W
 [
 f
 ]
 (
 x
 )
 
 
 {\displaystyle W[f](x)}
 
. Note that 
 
 
 
 F
 (
 x
 )
 
 
 {\displaystyle F(x)}
 
 need not exist for every real number 
 
 
 
 x
 
 
 {\displaystyle x}
 
, when the defining integral fails to converge.
The Weierstrass transform is intimately related to the <a href="/facts/Heat_equation/2JPPr5GG">heat equation</a> (or, equivalently, the <a href="/facts/Diffusion_equation/l3CUuLqF">diffusion equation</a> with constant diffusion coefficient). If the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 describes the initial temperature at each point of an infinitely long rod that has constant <a href="/facts/Thermal_conductivity/qWnSG3nQ">thermal conductivity</a> equal to 1, then the temperature distribution of the rod 
 
 
 
 t
 =
 1
 
 
 {\displaystyle t=1}
 
 time units later will be given by the function 
 
 
 
 F
 
 
 {\displaystyle F}
 
. By using values of 
 
 
 
 t
 
 
 {\displaystyle t}
 
 different from 1, we can define the generalized Weierstrass transform of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.
The generalized Weierstrass transform provides a means to approximate a given integrable function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 arbitrarily well with <a href="/facts/Analytic_function/JhL3z8FP">analytic functions</a>.

Weierstrass transform open-in-new

Weierstrass transform