论文标题
关于无限变化函数近似$ l^2 $复制内核希尔伯特空间的紧凑嵌入的简短说明
A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation
论文作者
论文摘要
该注释由两个很大程度上独立的部分组成。在第一部分中，我们给出了一个复制的内核hilbert space $ h $ h $ ncopedds $ l^2（ω，μ）的核心hilbert space $ h $持续添加到$ h $等同于$ h $ compact $ l^$ l^2的情况下， 在第二部分中，我们考虑了来自Infinite-Variate $ l^2 $ Approximation的情况。 假设将复制的单变量函数的繁殖内核希尔伯特空间嵌入，并将内核$ 1+k $嵌入$ l^2（ω，μ）$中是紧凑的。我们提供了一个简单的标准，用于检查重现的内核希尔伯特空间的嵌入紧凑性，并用$$ \ sum_ {u \ in \ Mathcal {u}}γ_U\ bigotimes_ | U | <\ infty \}，$和$（γ_U）_ {u \ in \ Mathcal {u}} $是一系列非阴性数字，以适当的$ l^2 $ space。
This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: Ω\times Ω\rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(Ω, μ),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(Ω, μ).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(Ω, μ)$ is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel Hilbert space with the kernel given by $$\sum_{u \in \mathcal{U}} γ_u \bigotimes_{j \in u}k,$$ where $\mathcal{U} = \{u \subset \mathbb{N}: |u| < \infty\},$ and $(γ_u)_{u \in \mathcal{U}}$ is a sequence of non-negative numbers, into an appropriate $L^2$ space.
