The characterization for (weak) nonhomogeneous sub-wavelet dual frames and their mixed oblique extension principle

In this paper, we investigate a class of weak dual frames generated by nonhomogeneous sub-wavelet systems on the half-line [Formula: see text]. Unlike classical wavelet frame theory on [Formula: see text] or [Formula: see text], the lack of a group structure on [Formula: see text] prevents the direct application of standard Fourier analysis. To address this, we work within the framework of the Walsh-Fourier transform associated with the operation “⊕”. We introduce a class of nonhomogeneous sub-wavelet systems and establish the Walsh-Fourier domain equation for their (weak) oblique dual frames in [Formula: see text]. Furthermore, we present a mixed oblique extension principle (MOEP) for such dual frames.