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Orthogonal wavelet

The scaling function is a refinable function. That is, it is a fractal functional equation, called the refinement equation (twin-scale relationordilation equation):

${\displaystyle \phi ($x)=\sum _{k=0}^{N-1}a_{k}\phi (2x-k)} ,

where the sequence ${\displaystyle (a_{0},\dots ,a_{N-1})}$ofreal numbers is called a scaling sequence or scaling mask. The wavelet proper is obtained by a similar linear combination,

${\displaystyle \psi ($x)=\sum _{k=0}^{M-1}b_{k}\phi (2x-k)} ,

where the sequence ${\displaystyle (b_{0},\dots ,b_{M-1})}$ 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:

${\displaystyle \sum _{n\in \mathbb {Z} }a_{n}a_{n+2m}=2\delta _{m,0}}$,

where ${\displaystyle \delta _{m,n}}$ is the Kronecker delta.

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 ${\displaystyle b_{n}=(-1)^{n}a_{N-1-n}}$. In some cases the opposite sign is chosen.

A necessary condition for the existence of a solution to the refinement equation is that there exists a positive integer A such that (see Z-transform):

${\displaystyle (1+Z)^{A}|a($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 Aof${\displaystyle \phi }$ corresponds to A vanishing moments of the dual wavelet ${\displaystyle {\tilde {\psi }}}$, that is, the scalar productsof${\displaystyle {\tilde {\psi }}}$ with any polynomial up to degree A-1 are zero. In the opposite direction, the approximation order Ãof${\displaystyle {\tilde {\phi }}}$ is equivalent to à vanishing moments of ${\displaystyle \psi }$. In the orthogonal case, A and à coincide.

A sufficient condition for the existence of a scaling function is the following: if one decomposes ${\displaystyle a($Z)=2^{1-A}(1+Z)^{A}p(Z)} , and the estimate

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

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

• Suppose ${\displaystyle p($Z)=1} then ${\displaystyle a($Z)=2^{1-A}(1+Z)^{A}} , and the estimate holds for n=A-2. The solutions are Schoenbergs B-splines of order A-1, where the (A-1)-th derivative is piecewise constant, thus the (A-2)-th derivative is Lipschitz-continuous. A=1 corresponds to the index function of the unit interval.

• A=2 and p linear may be written as

${\displaystyle a($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 ${\displaystyle c^{2}=3.}$ The positive root gives the scaling sequence of the D4-wavelet, see below.

• Ingrid Daubechies: Ten Lectures on Wavelets, SIAM 1992.
• Proc. 1st NJIT Symposium on Wavelets, Subbands and Transforms, April 1990.