Thm for Hilbert Spaces Reproducing Kernels References Deleted Scenes Reproducing Kernels [3] Suppose that (V, ⟨·, ·⟩) is Hilbert space of functions on Ω for which function evaluation is a bounded, linear functional. Then there exists, K : Ω × Ω → R called a reproducing kernel for which K(t, x) = K(x, t) symmetry , K(·, x) ∈ V belonging , f(x) = ⟨K(·, x), f⟩ reproduction ∀t, x ∈ Ω, f ∈ V Combining with the Riesz Representation Theorem ERR(f) := [0,1]d f(t) dt − 1 n n i=1 f(xi ) = ⟨η, f⟩ , representer η =? η(x) = reproduction ⟨K(·, x), η⟩ = symmetry ⟨η, K(·, x)⟩ = representer ERR K(·, x) = [0,1]d K(t, x) dt − 1 n n i=1 K(xi , x) ∥η∥2 = ⟨η, η⟩ = representer ERR(η) = [0,1]2d K(t, x) dt dx − 2 n n i=1 [0,1]d K(xi , x) dx + 1 n2 n i,j=1 K(xi , xj ) 8/17