论文标题
剪力结构变换的连续性特性和lizorkin型空间上的剪切合成算子
Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces
论文作者
论文摘要
我们为剪切式变换开发一个分配框架$ \ MATHCAL {s}_ψ\ COLON \ MATHCAL {S} _0（\ MATHBB {r}^2）\ to \ Mathcal {s} $ \ MATHCAL {s}^T_ψ\ Colon \ Mathcal {s}（\ Mathbb {s}）\ to \ Mathcal {S} _0（\ Mathbb {r}^2）$ $ \ mathcal {s}（\ Mathbb {s}）$是标准剪切组$ \ mathbb {s} $上高度局部测试功能的空间。这些空间及其双重$ \ MATHCAL {s} _0^\ prime（\ Mathbb r^2），\，\ Mathcal {s}^\ prime（\ Mathbb {s}）$称为测试功能和分布的lizorkin类型空间。当可接受的向量$ψ$属于$ \ Mathcal {s} _0（\ Mathbb {r}^2）$时，我们分析了这些变换的连续性属性。然后，我们分别定义了lizorkin型分布的剪切变换和剪切合成算子，分别是剪切合成算子和剪切型转换的转置映射。它们从$ \ Mathcal {s} _0^\ prime（\ Mathbb r^2）$转到$ \ Mathcal {s}^\ prime（\ Mathbb {s}）$，以及$ \ Mathcal {s}}^\ prime（Mathbb s） （\ Mathbb {r}^2）$。此外，我们显示了定义与剪切转换的一致性，该剪切变换通过直接评估剪切物上的分布来定义。剪切合成算子也可以做同样的事情。最后，我们给出了lizorkin型分布的重建公式，从中可以将这种广义功能的作用写成与标准剪切组相比的绝对收敛组成部分。
We develop a distributional framework for the shearlet transform $\mathcal{S}_ψ\colon\mathcal{S}_0(\mathbb{R}^2)\to\mathcal{S}(\mathbb{S})$ and the shearlet synthesis operator $\mathcal{S}^t_ψ\colon\mathcal{S}(\mathbb{S})\to\mathcal{S}_0(\mathbb{R}^2)$, where $\mathcal{S}_0(\mathbb{R}^2)$ is the Lizorkin test function space and $\mathcal{S}(\mathbb{S})$ is the space of highly localized test functions on the standard shearlet group $\mathbb{S}$. These spaces and their duals $\mathcal{S}_0^\prime (\mathbb R^2),\, \mathcal{S}^\prime (\mathbb{S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector $ψ$ belongs to $\mathcal{S}_0(\mathbb{R}^2)$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from $\mathcal{S}_0^\prime (\mathbb R^2)$ to $\mathcal{S}^\prime (\mathbb{S})$ and from $\mathcal{S}^\prime (\mathbb S)$ to $\mathcal{S}_0^\prime (\mathbb{R}^2)$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.
