$(p,q)$-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}(Φ)=\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})ϕ_{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 $\{ϕ_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in\mathbf{Z}\times\mathbf{Z}^d,1\leq i\leq r\}$ to constitute a $(p,q)$-frame of $V_{p,q}(Φ)$ are given. Moreover, the result shows that $V_{p,q}(Φ)$ is closed under these equivalent conditions of $(p,q)$-frame for the family $\{ϕ_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in\mathbf{Z}\times\mathbf{Z}^d,1\leq i\leq r\}$, although the general result is not correct.

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