Three types of reproducing kernel Hilbert spaces of polynomials

Abstract

In this paper we will investigate reproducing kernel Hilbert spaces of polynomials of degree at most n with three different inner products: given by an integral with a weight, given by the sum of products of values of a polynomial at n + 1 points and given by the sum of products of coefficients of the same power. In the first case we will show that the reproducing kernel depends continuously on deformation of an inner product in a precisely defined sense. In the second and third case we will give a formula for the reproducing kernel.