Frames associated with shift invariant spaces on positive half line

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L²(ℝ⁺). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L²(ℝ⁺). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.