Walsh Shift-Invariant Sequences and p-adic Nonhomogeneous Dual Wavelet Frames in $$L^{2}({\mathbb R}_{+})$$

Abstract

It is known from the existing literatures that nonhomogeneous wavelet systems play a fundamental role in wavelet analysis which benefit to understanding many aspects of wavelet theory and relate to different aspects of wavelet analysis. Now the nonhomogeneous dual wavelet frames in $$L^{2}(\mathbb R)$$ have been extensively studied, while the ones in $$L^{2}(\mathbb R_{+})$$ are not. This is also true for shift-invariant sequences in $$L^{2}(\mathbb {R}_{+})$$ . Intuitively, $$L^{2}(\mathbb {R}_{+})$$ -wavelet frames can be obtained by projection from $$L^{2}(\mathbb R)$$ -ones, while it is not the case for $$L^{2}(\mathbb {R}_{+})$$ since the projections do not have complete affine structure. This is partially because $$\mathbb R_{+}$$ is not a group in terms of usual addition. It is worth noting that $$\mathbb R_{+}$$ is a group according to the operation “ $$\oplus $$ ” by which the Walsh–Fourier transform is defined. Using Walsh–Fourier transform method, we in this paper characterize the shift-invariant Bessel sequences, frame sequences and Riesz sequences in $$L^{2}(\mathbb {R}_{+})$$ ; and give a characterization of p-adic nonhomogeneous dual wavelet frames in $$L^{2}(\mathbb {R}_{+})$$ .