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

Abstract

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 \(\{\phi_{i}(\cdot-j_{1},\cdot-j_{2}):(j_{1},j_{2})\in\mathbf{Z}\times\mathbf{Z}^d,1\leq i\leq r\}\) to constitute a \((p,q)\)-frame of \(V_{p,q}(\Phi)\) are given. Moreover, the result shows that \(V_{p,q}(\Phi)\) 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.