$(p,q)$-frames in shift-invariant subspaces of mixed Lebesgue spaces $L^{p,q}(\mathbf{R}\times {\mathbf{R}}^d)$

In this paper, we mainly discuss the $(p,q)$-frame in shift-invariant subspace \begin{equation*} V_{p,q}(\Phi)=\left{\sum\limits_{i=1}^{r}\sum\limits_{j_{1}\in $\mathbf{Z}$}\sum\limits_{j_{2}\in $\mathbf{Z}$^{d}}d_{i}(j_{1},j_{2})\phi_{i}(\cdot-j_{1},\cdot-j_{2}):\Big(d_{i}(j_{1},j_{2})\Big){(j{1},j_{2})\in $\mathbf{Z}$\times$\mathbf{Z}$^{d}}\in \ell^{p,q}($\mathbf{Z}$\times$\mathbf{Z}$^d)\right} \end{equation*} of mixed Lebesgue space $L^{p,q}(\mathbf{R}\times {\mathbf{R}}^d)$. Some equivalent conditions for ${\phi_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in$\mathbf{Z}$\times$\mathbf{Z}$^d,1\leq i\leq r}$toconstitutea$(p,q)$-frameof$V_{p,q}(\Phi)$ are given. Moreover, the result shows that $V_{p,q}(\mathrm{\Phi })$ is closed under these equivalent conditions of $(p,q)$-frame for the family ${\phi_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in$\mathbf{Z}$\times$\mathbf{Z}$^d,1\leq i\leq r}$, although the general result is not correct.