Given a Banach space $E$ consisting of functions, we ask whether there exists a reproducing kernel Hilbert space $H$ with bounded kernel such that $E\subset H$. More generally, we consider the question, whether for a given Banach space consisting of functions $F$ with $E\subset F$, there exists an intermediate reproducing kernel Hilbert space $E\subset H\subset F$. We provide both sufficient and necessary conditions for this to hold. Moreover, we show that for typical classes of function spaces described by smoothness there is a strong dependence on the underlying dimension: the smoothness $s$ required for the space $E$ needs to grow \emph{proportional} to the dimension $d$ in order to allow for an intermediate reproducing kernel Hilbert space $H$.

翻译：给定一个由函数构成的巴拿赫空间$E$，我们询问是否存在一个有界核的可再生核希尔伯特空间$H$，使得$E\subset H$。更一般地，我们考虑如下问题：对于给定的由函数构成且满足$E\subset F$的巴拿赫空间$F$，是否存在一个中间的可再生核希尔伯特空间$E\subset H\subset F$？我们给出了这一性质成立的充分条件和必要条件。此外，我们证明，对于由光滑性描述的典型函数空间类，其成立性强烈依赖于底维数：为了使中间的可再生核希尔伯特空间$H$存在，空间$E$所需的光滑性$s$必须随维数$d$成比例增长。