$(p,q)$-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}(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 $\{{\varphi }_i(\cdot -j_1,\cdot -j_2):(j_1,j_2)\in \mathbf{Z}\times {\mathbf{Z}}^d,1\le i\le r\}$ 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 $\{{\varphi }_i(\cdot -j_1,\cdot -j_2):(j_1,j_2)\in \mathbf{Z}\times {\mathbf{Z}}^d,1\le i\le r\}$, although the general result is not correct.

References in corpus (1)

• Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed Lebesgue space