Abstract
In this paper, we consider the closedness of shift invariant subspaces in Lp,q(Rd+1) . We first define the shift invariant subspaces generated by the shifts of finite functions in Lp,q(Rd+1) . Then we give some necessary and sufficient conditions for the shift invariant subspaces in Lp,q(Rd+1) to be closed. Our results improve some known results in (Aldroubi et al. in J. Fourier Anal. Appl. 7:1-21, 2001).