The structure of finitely generated shift-invariant spaces in mixed Lebesgue spaces $$L^{p,q}\left( {\mathbb {R}}^{d+1}\right) $$Lp,qRd+1

In this paper, we investigate the structure of finitely generated shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$. We characterize shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ in terms of the semi-discrete convolutions of their generators with sequences in $$\ell ^{p,q}\left( \mathbb {Z}^{d+1}\right) $$. In addition, if the generators of shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ are compactly supported, we prove that shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ are the intersection of $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ and the linear space of functions formed with the semi-discrete convolutions of their generators with sequences on $$\mathbb {Z}^{d+1}$$.