In this paper, we study nonhomogeneous wavelet systems which have close
relations to the fast wavelet transform and homogeneous wavelet systems. We
introduce and characterize a pair of frequency-based nonhomogeneous dual
wavelet frames in the distribution space; the proposed notion enables us to
completely separate the perfect reconstruction property of a wavelet system
from its stability property in function spaces. The results in this paper lead
to a natural explanation for the oblique extension principle, which has been
widely used to construct dual wavelet frames from refinable functions, without
any a priori condition on the generating wavelet functions and refinable
functions. A nonhomogeneous wavelet system, which is not necessarily derived
from refinable functions via a multiresolution analysis, not only has a natural
multiresolution-like structure that is closely linked to the fast wavelet
transform, but also plays a basic role in understanding many aspects of wavelet
theory. To illustrate the flexibility and generality of the approach in this
paper, we further extend our results to nonstationary wavelets with real
dilation factors and to nonstationary wavelet filter banks having the perfect
reconstruction property