Walsh Shift-Invariant Sequences and p-adic Nonhomogeneous Dual Wavelet Frames in L 2 (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 (R) have been extensively studied, while the ones in L 2 (R + ) are not. This is also true for shift-invariant sequences in L 2 (R + ). Intuitively, L 2 (R + ) -wavelet frames can be obtained by projection from L 2 (R) -ones, while it is not the case for L 2 (R + ) since the projections do not have complete affine structure. This is partially because R + is not a group in terms of usual addition. It is worth noting that R + is a group according to the operation “⊕ ” 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 (R + ) ; and give a characterization of p-adic nonhomogeneous dual wavelet frames in L 2 (R + ).