论文标题
$（p，q）$ - 混合Lebesgue Spaces的shift-invariant子空间$ l^{p，q}（\ mathbf {r} \ times \ times \ mathbf {r}^{d} {d}）$
$(p,q)$-frames in shift-invariant subspaces of mixed Lebesgue spaces $L^{p,q}(\mathbf{R}\times \mathbf{R}^{d})$
论文作者
论文摘要
在本文中，我们主要讨论$（p，q）$ - shift-invariant子空间\ begin {qore*} v_ {p，q}（φ）= \ left \ sum \ sum \ limits_ {i = 1} \ Mathbf {Z}} \ sum \ limits_ {J_ {2} \ in \ Mathbf {z}^{d}} d_ {i}（j_ {1}，j_ {2}）ϕ__ {i}（\ cdot-j_ {1}，\ cdo t-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 \ times \ times \ mathbf {z}^d）\ right \} \ right \} $ l^{p，q}（\ mathbf {r} \ times \ mathbf {r}^{d}）$。 $ \ {ϕ_ {i}（\ cdot-j_ {1}，\ cdot-j_ {2}）的一些等效条件$（p，q）$ - $ v_ {p，q}（φ）$的帧。此外，结果表明$ v_ {p，q}（φ）$在$（p，q）$ - 家族的这些等效条件下关闭$ \ {ϕ_ {i}（\ cdot-j_ {1}，\ cdot-j_ {2}）:(
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.
