论文标题
Lizorkin分布的Banach子空间的明确表示
Explicit representations for Banach subspaces of Lizorkin distributions
论文作者
论文摘要
Lizorkin空间非常适合研究各种操作员。例如，分数laplacians和ra的变换。在本文中，我们表明，不幸的是，在施瓦茨空间中没有补充该空间。但是，我们可以证明它在$ C_0（\ Mathbb r^d）$中是密集的，这是一个由较大的Schwartz空间共享的属性，事实证明这对于应用程序很有用。基于此结果，我们研究了Banach空间的Lizorkin分布的子空间，并为之持续代表操作员。然后，我们引入了一个涉及这些空间的变异框架，并利用构造的运算符。通过研究此框架的两个特定案例，我们能够加强分数花纹和2层Relu网络的现有结果。
The Lizorkin space is well-suited for studying various operators; e.g., fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. However, we can show that it is dense in $C_0(\mathbb R^d)$, a property that is shared by the larger Schwartz space and that turns out to be useful for applications. Based on this result, we investigate subspaces of Lizorkin distributions that are Banach spaces and for which a continuous representation operator exists. Then, we introduce a variational framework involving these spaces and that makes use of the constructed operator. By investigating two particular cases of this framework, we are able to strengthen existing results for fractional splines and 2-layer ReLU networks.
