(Existence
of biorthogonal compactly supported wavelets ) There exist smooth compactly
supported real valued biorthogonal wavelets
calculated according to the following procedure:

1. Pick a number
.
It will influence the length of the scaling filters
and the size of support of
,
see the proposition (
Support of
scaling function
).

There is more then one way to achieve a match of the condition 4. One way to
do it is to select a scaling function
as in the proposition (
Existence
of biorthogonal basis 1
) (using the proposition
(
Shifted Fourier transform
equality
) to verify applicability), calculate the corresponding
as in the proposition (
Scaling equation 4
)
and then seek
to satisfy the condition 4. The following sections contain examples.