A sufficient condition for n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces

Abstract

We show that if a reproducing kernel Hilbert space H K , consisting of functions defined on E, enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number n, and any function f ∈ H K , there exists a set of n parameterized multiple kernels Kw1 ,By applying the theorem of this paper we show that the Hardy space and the Bergman space, as well as all the weighted Bergman spaces in the unit disc all possess n-best approximations. In the Hardy space case this gives a new proof of a classical result. Based on the obtained results we further prove existence of n-best spherical Poisson kernel approximation to functions of finite energy on the real-spheres.