Which Spaces can be Embedded in Reproducing Kernel Hilbert Spaces?

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Abstract Given a Banach space E consisting of functions, we ask whether there exists a reproducing kernel Hilbert space H with bounded kernel such that $$E\subset H$$ E ⊂ H . More generally, we consider the question, whether for a given Banach space consisting of functions F with $$E\subset F$$ E ⊂ F , there exists an intermediate reproducing kernel Hilbert space $$E\subset H\subset F$$ E ⊂ H ⊂ 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 proportional to the dimension d in order to allow for an intermediate reproducing kernel Hilbert space H.