Stability of Biorthogonal Wavelet Bases

We investigate the lifting scheme as a method for constructing compactly supported biorthogonal scaling functions and wavelets. A well-known issue arising with the use of this scheme is that the resulting functions are only formally biorthogonal. It is not guaranteed that the new wavelet bases actually exist in an acceptable sense. To verify that these bases do exist one must test an associated linear operator to ensure that it has a simple eigenvalue at one and that all its remaining eigenvalues have modulus less than one, a task which becomes numerically intensive if undertaken repeatedly. We simplify this verification procedure in two ways: (i) we show that one need only test an identifiable half of the eigenvalues of the operator, (ii) we show that when the operator depends upon a single parameter, the test first fails for values of that parameter at which the eigenvalue at one becomes a multiple eigenvalue. We propose that this new verification procedure comprises a first step towards determining simple conditions, supplementary to the lifting scheme, to ensure existence of the new wavelets generated by the scheme and develop an algorithm to this effect.