Numerical Stability of Biorthogonal Wavelet Transforms

. For orthogonal wavelets, the discrete wavelet and wave packet transforms and their inverses are orthogonal operators with perfect numerical stability. For biorthogonal wavelets, numerical instabilities can occur. We derive bounds for the 2-norm and average 2-norm of these transforms, including efficient numerical estimates if the number L of decomposition levels is small, as well as growth estimates for L !1. These estimates allow easy determination of numerical stability directly from the wavelet coefficients. Examples show that many biorthogonal wavelets are in fact numerically well behaved. 1. Introduction The discrete wavelet transform and wave packet transform have become well established in many applications, such as signal processing. Originally derived for orthogonal wavelets, they can equally well be based on biorthogonal wavelets. Numerical experiments with various types of biorthogonal wavelets show that in some cases considerable roundoff error is accumulated during deco..