A unified theory of radial basis functions. Native Hilbert spaces for radial basis functions. II
[For part I see ISNM, Int. Ser. Numer. Math. 132, 255-282 (1999; Zbl 0944.46017).] Given some symmetric positive definite function \(\Phi\) on some domain of Euclidean space, one can define the native Hilbert space as the completion of the linear hull of the functions \(\Phi(x,\cdot)\) with respect to the inner product \((\Phi(x,\cdot),\Phi(y,\cdot))_\Phi= \Phi(x,y)\) (as a model serves the case of radial basis functions \(\Phi(x,y)=\phi(\|x-y\|)\)). Under a mild boundedness condition, the native Hilbert space can be embedded into \(L_2\), and the adjoint of this embedding is the integral operator with kernel \(\Phi\). Using eigenfunction expansion for this operator in \(L_2\), a different characterization of the native space is obtained. Also some convergence results are given.
scientific article; zbMATH DE number 1533133
scientific article