Computing a family of reproducing kernels for statistical applications

Abstract

For an open subset Ω of ℝ, an integer,m, and a positive real parameter τ, the Sobolev spacesH ^m (Ω) equipped with the norms: ∥u∥²=∫u(t)²dt+(1/τ^2m∫u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When Ω is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.