论文标题
两个运营商的轨道框架
Frames by orbits of two operators that commute
论文作者
论文摘要
通过有限操作员的迭代，由向量轨道形成的框架最近引起了广泛的关注，尤其是由于其应用于动力学采样。在本文中，我们考虑了两个在某些可分开的希尔伯特空间上行动的有界运营商$ \ mathcal h $。我们完全表征运算符$ t $和$ l $，带有$ tl = lt $，并设置$φ\ subset \ mathcal h $，使得集合$ \ {t^k l^j ϕ：k in \ in \ mathbb z，in j，in j，indbb z \ in j，dement in q inφ\} $构成$ \ m rath h $ \ mathcal h $的框架。这是根据在圆环上定义的平方集成函数空间的模型子空间来完成的，并且在某些耐寒空间中具有多重性。作用于这些模型的操作员是双侧转移和单侧转移的压缩（表现尖锐）。此上下文包括当Hilbert Space $ \ Mathcal H $是$ l^2（\ Mathbb r）$的子空间（在整数沿翻译下不变的子空间），其中操作员$ t $是一个和$ l $的翻译，是一个转移的运算符。
Frames formed by orbits of vectors through the iteration of a bounded operator have recently attracted considerable attention, in particular due to its applications to dynamical sampling. In this article, we consider two commuting bounded operators acting on some separable Hilbert space $\mathcal H$. We completely characterize operators $T$ and $L$ with $TL=LT$ and sets $Φ\subset \mathcal H$ such that the collection $\{T^k L^j ϕ: k\in \mathbb Z, j\in J, ϕ\in Φ\}$ forms a frame of $\mathcal H$. This is done in terms of model subspaces of the space of square integrable functions defined on the torus and having values in some Hardy space with multiplicity. The operators acting on these models are the bilateral shift and the compression of the unilateral shift (acting pointwisely). This context includes the case when the Hilbert space $\mathcal H$ is a subspace of $L^2(\mathbb R)$, invariant under translations along the integers, where the operator $T$ is the translation by one and $L$ is a shift-preserving operator.
