Refinable function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, in the area of <a href="/facts/Wavelet/3XcX4uIP">wavelet</a> analysis, a refinable function is a function which fulfils some kind of <a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a>. A function 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 is called refinable with respect to the mask 
 
 
 
 h
 
 
 {\displaystyle h}
 
 if

φ
        (
        x
        )
        =
        2
        ⋅
        
          ∑
          
            k
            =
            0
          
          
            N
            −
            1
          
        
        
          h
          
            k
          
        
        ⋅
        φ
        (
        2
        ⋅
        x
        −
        k
        )
      
    
    {\displaystyle \varphi (x)=2\cdot \sum _{k=0}^{N-1}h_{k}\cdot \varphi (2\cdot x-k)}

This condition is called refinement equation, dilation equation or two-scale equation.
Using the <a href="/facts/Convolution/PVPrdz9J">convolution</a> (denoted by a star, *) of a function with a discrete mask and the dilation operator 
 
 
 
 D
 
 
 {\displaystyle D}
 
 one can write more concisely:

φ
        =
        2
        ⋅
        
          D
          
            1
            
              /
            
            2
          
        
        (
        h
        ∗
        φ
        )
      
    
    {\displaystyle \varphi =2\cdot D_{1/2}(h*\varphi )}

It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back.
There is a similarity to <a href="/facts/Iterated_function_systems/j56g6yEM">iterated function systems</a> and <a href="/facts/De_Rham_curve/IhaRey4N">de Rham curves</a>.
The operator 
 
 
 
 φ
 ↦
 2
 ⋅
 
 D
 
 1
 
 /
 
 2
 
 
 (
 h
 ∗
 φ
 )
 
 
 {\displaystyle \varphi \mapsto 2\cdot D_{1/2}(h*\varphi )}
 
 is linear.
A refinable function is an <a href="/facts/Eigenfunction/62EJS0XB">eigenfunction</a> of that operator.
Its absolute value is not uniquely defined.
That is, if 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 is a refinable function,
then for every 
 
 
 
 c
 
 
 {\displaystyle c}
 
 the function 
 
 
 
 c
 ⋅
 φ
 
 
 {\displaystyle c\cdot \varphi }
 
 is refinable, too.
These functions play a fundamental role in <a href="/facts/Wavelet/3XcX4uIP">wavelet</a> theory as <a href="/facts/Wavelet/3XcX4uIP">scaling functions</a>.

Refinable function open-in-new

Refinable function