Orthogonal wavelet
An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened one may end up with biorthogonal wavelets. Basics. The scaling function is a refinable function. That is, it is a fractal functional equation, called the refinement equation (twin-scale relation or dilation equation): where the sequence formula_2 of real numbers is called a scaling sequence or scaling mask. The wavelet proper is obtained by a similar linear combination, where the sequence formula_4 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: where formula_6 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 formula_7. In some cases the opposite sign is chosen. Vanishing moments, polynomial approximation and smoothness. 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. 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 formula_9 corresponds to "A" vanishing moments of the dual wavelet formula_10, that is, the scalar products of formula_10 with any polynomial up to degree "A-1" are zero. In the opposite direction, the approximation order "Ã" of formula_12 is equivalent to "Ã" vanishing moments of  formula_13. In the orthogonal case, "A" and "Ã" coincide. A sufficient condition for the existence of a scaling function is the following: if one decomposes formula_14, and the estimate holds for some formula_16, then the refinement equation has a "n" times continuously differentiable solution with compact support.