Background Riesz Rep Thm in R2 Inner Product Riesz Rep 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