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M to be linearly independent over G. This implies α = 0 and we are left with P β = 0. 14, EQsys3). 4, ORP) if the M × q matrix P with entries λj (pi ), 1 ≤ j ≤ M, 1 ≤ i ≤ q is injective. This condition means that the only element p ∈ P with vanishing data λ1 (p), . . , λM (p) must be the zero element. 14, EQsys3) and how a conditionally positive definite function Φ can arise in the above Hilbert space setting. 1 Optimal Recovery in Hilbert Spaces 35 generalized conditionally positive definite function with P generalizing IPmd by Φ(x, y) := (gx , gy )G , x, y ∈ Ω.

9, Prq). We use this to go over to a Lagrange-type basis of P with respect to Ξ which we again denote by p1 , . . , pq . Then P is the identity matrix and we can write every function p ∈ P as (PRq2) q p(·) = p(ξj )pj (·). 10) j=1 This now yields the explicit form of a projector ΠP onto P as q ΠP (f )(·) := f (ξj )pj (·) j=1 for all functions that are at least defined on Ξ. The projector has the additional property (f − ΠP f )(Ξ) = {0} for all functions f that are defined on Ξ, because of δξj ,Ξ = 0, 1 ≤ j ≤ q.

7, lfgeneral): (lf2) f (x) := (δx,Ξ , F −1 (f ))Φ , x ∈ Ω, f ∈ F. 13) This assigns specific function values to the abstract element of the closure F of F0 . The assignment has the consequence that f (Ξ) = {0} due to δξj ,Ξ = 0, 1 ≤ j ≤ q, and thus it is rather an assignment of values to f − ΠP f than to f itself. We thus avoid this complication and define a mapping RP : F → G by (lf3) (RP f )(x) := (δx,Ξ , F −1 (f ))Φ , x ∈ Ω, f ∈ F. 3 Native Spaces 57 We have to show that this maps into G, and for this we have to evaluate M αj (δxj ,Ξ , F −1 (f ))Φ λX,M,α (RP (f )) = j=1  M =  j=1  αj δxj ,Ξ , F −1 (f ) .

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