The closedness of shift invariant subspaces in L p , q ( R d + 1 ) $L^{p,q} (\mathbb{R}^{d+1} )$

In this paper, we consider the closedness of shift invariant subspaces in $L^{p,q} (\mathbb{R}^{d+1} )$ . We first define the shift invariant subspaces generated by the shifts of finite functions in $L^{p,q} (\mathbb{R}^{d+1} )$ . Then we give some necessary and sufficient conditions for the shift invariant subspaces in $L^{p,q} (\mathbb{R}^{d+1} )$ to be closed. Our results improve some known results in (Aldroubi et al. in J. Fourier Anal. Appl. 7:1–21, 2001).