Functional Reproducing Kernel Spaces Based On Data From Interpolation Error

When processing point evaluation data,one necessary requirement is that the sampling process should be stable.This gives rise to the notion of reproducing kernel Hilbert spaces(RKHSs).Due to the guarantee that point evaluation functionals are continuous on RKHSs,RKHSs have been an ideal choice to handle point evaluation data.However,there are a large number of occasions,such as signal processing or X-ray scanning,when the data available are usually measured through some linear functionals other than through point evaluation functionals.In order for the stability of such sampling processes,the notion of functional reproducing kernel spaces(FRKSs)have been put forward.Since such spaces can ensure the stability of a family of linear functionals,FRKSs become an ideal choice when dealing with data measured through the corresponding linear functionals.By introducing a notion of the interpolation error functionals,we bring about a specific family of FRKSs,namely,the FRKSs based on interpolation error functionals.After that,we show that such FRKSs are not perfect FRKSs.As a result of the continuity of interpolation error functionals on such spaces,we use both the minimum norm interpolation model and the regularization model on such FRKSs to deal with data from interpolation error.Aside from those,due to the special structure of the interpolation error functionals,we show that the FRKSs based on interpolation error functionals and the classic conditionally positive definite kernel spaces are tightly connected to each other.We also use the minimum norm interpolation model and the regularization model on both spaces to deal with point evaluation data.Finally,we show in a numerical means that the FRKSs based on interpolation error functionals perform well on point evaluation data.