A short note on compact embeddings of reproducing kernel Hilbert spaces in L^2 for infinite-variate function approximation
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This note consists of two largely independent parts. In the first part we give conditions on the kernel k: Ω×Ω→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 ∑_u ∈𝒰γ_u ⊗_j ∈ uk, where 𝒰 = {u ⊂ℕ: |u| < ∞}, and (γ_u)_u ∈𝒰 is a sequence of non-negative numbers, into an appropriate L^2 space.
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