Regularization method for the generalized moment problem in a functional reproducing kernel hilbert space

Functional reproducing kernel Hilbert spaces (FRKHSs) are appropriate function spaces in which we seek a target function from a finite number of non-point-evaluation functional data. We consider reconstructing a function from a finite number of generalized moment data via regularization in an FRKHS with respect to the generalized moment functionals. We construct specific FRKHSs and their associated FRKHS kernels with respect to two classes of generalized moment functionals, the Hamburger moment functionals and the trigonometric moment functionals. We solve the regularization problem in the resulting FRKHSs by the representer theorem. Numerical examples are presented to illustrate the better performance of regularization in an FRKHS than regularization in the square integrable functions space.