If this condition is weakened one may end up with biorthogonal wavelets.
The wavelet proper is obtained by a similar linear combination, where the sequence
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: where
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
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): The maximally possible power A is called polynomial approximation order (or pol.
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
corresponds to A vanishing moments of the dual wavelet
In the opposite direction, the approximation order à of
is equivalent to à vanishing moments of
In the orthogonal case, A and à coincide.
A sufficient condition for the existence of a scaling function is the following: if one decomposes
, then the refinement equation has a n times continuously differentiable solution with compact support.