Orthogonal wavelet

<h2 id="basics">Basics</h2>
The <a href="/facts/Wavelet/3XcX4uIP">scaling function</a> is a <a href="/facts/Refinable_function/K8G1hMEk">refinable function</a>.
That is, it is a <a href="/facts/Fractal/sXj38JeN">fractal</a> <a href="/facts/Functional_equation/4ovU4CAt">functional equation</a>, called the refinement equation (twin-scale relation or dilation equation):

ϕ
 (
 x
 )
 =
 
 ∑
 
 k
 =
 0
 
 
 N
 −
 1
 
 
 
 a
 
 k
 
 
 ϕ
 (
 2
 x
 −
 k
 )
 
 
 {\displaystyle \phi (x)=\sum _{k=0}^{N-1}a_{k}\phi (2x-k)}
 
,
where the sequence 
 
 
 
 (
 
 a
 
 0
 
 
 ,
 …
 ,
 
 a
 
 N
 −
 1
 
 
 )
 
 
 {\displaystyle (a_{0},\dots ,a_{N-1})}
 
 of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> is called a scaling sequence or scaling mask.
The wavelet proper is obtained by a similar linear combination,

ψ
 (
 x
 )
 =
 
 ∑
 
 k
 =
 0
 
 
 M
 −
 1
 
 
 
 b
 
 k
 
 
 ϕ
 (
 2
 x
 −
 k
 )
 
 
 {\displaystyle \psi (x)=\sum _{k=0}^{M-1}b_{k}\phi (2x-k)}
 
,
where the sequence 
 
 
 
 (
 
 b
 
 0
 
 
 ,
 …
 ,
 
 b
 
 M
 −
 1
 
 
 )
 
 
 {\displaystyle (b_{0},\dots ,b_{M-1})}
 
 of real numbers is called a wavelet sequence or wavelet mask.
A necessary condition for the orthogonality of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients:

∑
 
 n
 ∈
 
 Z
 
 
 
 
 a
 
 n
 
 
 
 a
 
 n
 +
 2
 m
 
 
 =
 2
 
 δ
 
 m
 ,
 0
 
 
 
 
 {\displaystyle \sum _{n\in \mathbb {Z} }a_{n}a_{n+2m}=2\delta _{m,0}}
 
,
where 
 
 
 
 
 δ
 
 m
 ,
 n
 
 
 
 
 {\displaystyle \delta _{m,n}}
 
 is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>.
In this case there is the same number M=N of coefficients in the scaling as in the wavelet sequence, the wavelet sequence can be determined as 
 
 
 
 
 b
 
 n
 
 
 =
 (
 −
 1
 
 )
 
 n
 
 
 
 a
 
 N
 −
 1
 −
 n
 
 
 
 
 {\displaystyle b_{n}=(-1)^{n}a_{N-1-n}}
 
. In some cases the opposite sign is chosen.

<h2 id="vanishing-moments-polynomial-approximation-and-smoothness">Vanishing moments, polynomial approximation and smoothness</h2>
A necessary condition for the existence of a solution to the refinement equation is that there exists a positive integer A such that (see <a href="/facts/Z-transform/ERNvDJqt">Z-transform</a>):

(
        1
        +
        Z
        
          )
          
            A
          
        
        
          |
        
        a
        (
        Z
        )
        :=
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        Z
        +
        ⋯
        +
        
          a
          
            N
            −
            1
          
        
        
          Z
          
            N
            −
            1
          
        
        .
      
    
    {\displaystyle (1+Z)^{A}|a(Z):=a_{0}+a_{1}Z+\dots +a_{N-1}Z^{N-1}.}

The maximally possible power A is called polynomial approximation order (or pol. app. power) or number of vanishing moments. It describes the ability to represent polynomials up to degree A-1 with linear combinations of integer translates of the scaling function. 
In the biorthogonal case, an approximation order A of 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 corresponds to A vanishing moments of the dual wavelet 
 
 
 
 
 
 
 ψ
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\psi }}}
 
, that is, the <a href="/facts/Dot_product/tNz8MLjT">scalar products</a> of 
 
 
 
 
 
 
 ψ
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\psi }}}
 
 with any polynomial up to degree A-1 are zero. In the opposite direction, the approximation order Ã of 
 
 
 
 
 
 
 ϕ
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\phi }}}
 
 is equivalent to Ã vanishing moments of 
 
 
 
 ψ
 
 
 {\displaystyle \psi }
 
. In the orthogonal case, A and Ã coincide.
A sufficient condition for the existence of a scaling function is the following: if one decomposes 
 
 
 
 a
 (
 Z
 )
 =
 
 2
 
 1
 −
 A
 
 
 (
 1
 +
 Z
 
 )
 
 A
 
 
 p
 (
 Z
 )
 
 
 {\displaystyle a(Z)=2^{1-A}(1+Z)^{A}p(Z)}
 
, and the estimate

1
 ≤
 
 sup
 
 t
 ∈
 [
 0
 ,
 2
 π
 ]
 
 
 
 |
 
 p
 (
 
 e
 
 i
 t
 
 
 )
 
 |
 
 <
 
 2
 
 A
 −
 1
 −
 n
 
 
 ,
 
 
 {\displaystyle 1\leq \sup _{t\in [0,2\pi ]}\left|p(e^{it})\right|<2^{A-1-n},}

holds for some 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, then the refinement equation has a n times continuously differentiable solution with compact support.

<h3>Examples</h3>
<ul><li>Suppose 
 
 
 
 p
 (
 Z
 )
 =
 1
 
 
 {\displaystyle p(Z)=1}
 
 then 
 
 
 
 a
 (
 Z
 )
 =
 
 2
 
 1
 −
 A
 
 
 (
 1
 +
 Z
 
 )
 
 A
 
 
 
 
 {\displaystyle a(Z)=2^{1-A}(1+Z)^{A}}
 
, and the estimate holds for n=A-2. The solutions are Schoenbergs <a href="/facts/B-spline/MLTCWCEd">B-splines</a> of order A-1, where the (A-1)-th derivative is piecewise constant, thus the (A-2)-th derivative is <a href="/facts/Lipschitz_continuity/Hw20EPEU">Lipschitz-continuous</a>. A=1 corresponds to the index function of the unit interval.</li></ul>
<ul><li>A=2 and p linear may be written as</li></ul>

a
        (
        Z
        )
        =
        
          
            1
            4
          
        
        (
        1
        +
        Z
        
          )
          
            2
          
        
        (
        (
        1
        +
        Z
        )
        +
        c
        (
        1
        −
        Z
        )
        )
        .
      
    
    {\displaystyle a(Z)={\frac {1}{4}}(1+Z)^{2}((1+Z)+c(1-Z)).}

Expansion of this degree 3 polynomial and insertion of the 4 coefficients into the orthogonality condition results in 
  
    
      
        
          c
          
            2
          
        
        =
        3.
      
    
    {\displaystyle c^{2}=3.}
  
 The positive root gives the scaling sequence of the D4-wavelet, see below.

<ul><li><a href="/facts/Ingrid_Daubechies/vjRQuWwo">Ingrid Daubechies</a>: Ten Lectures on Wavelets, SIAM 1992.</li>
<li><a href="http://web.njit.edu/~akansu/s1.htm">Proc. 1st NJIT Symposium on Wavelets, Subbands and Transforms, April 1990.</a></li></ul>

Orthogonal wavelet open-in-new

Orthogonal wavelet