Weak Nonhomogeneous Wavelet Bi-Frames for Reducing Subspaces of Sobolev Spaces

As homogeneous wavelet bi-frames derived from a pair of refinable functions, nonhomogeneous wavelet bi-frames admit fast wavelet transform, which is one of main concerns in wavelet analysis. Simultaneously, for a nonhomogeneous wavelet bi-frame in a pair of dual spaces with s≠0, smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in . However, the requirement of the Bessel property of affine systems is always needed. For refinable-function based wavelet bi-frames, such a requirement is imposed on refinable functions in different ways. This is not what we expect. So a natural question involves what we are expecting from a pair of general refinable functions. For this purpose, we introduce the notion of the reducing subspace of a Sobolev space and the notion of the weak nonhomogeneous wavelet bi-frame in a general pair of dual reducing subspaces of . We characterize reducing subspaces of a Sobolev space. Then, under the setting of reducing subspaces, we study the properties of weak nonhomogeneous wavelet bi-frames, obtain a construction of weak nonhomogeneous wavelet bi-frames from a pair of refinable functions, and derive a fast wavelet algorithm associated with such weak nonhomogeneous wavelet bi-frames.