Computational Aspects

Abstract

In many applications, the choice of Hilbert space and norm is governed by context related modeling reasons and one has to face the problem of computing the corresponding reproducing kernel. Symmetrically, it is of interest to characterize the Hilbert space H _K associated with a given kernel K by the Moore-Aronszajn theorem and in particular to give necessary and sufficient conditions for a function to belong to 1iK. Gu and Wahba (1992) say: “T he norm and t he reproducing kernel in a RKHS determine each other uniquely, but like other duals in mathematical structures, the interpretability, and the availability of an explicit form for one part is often at the expense of the same for the other part”. Gu (2000) argues that it can be viewed as an inversion problem: “Just as the inverse J+ of a matrix J can rarely be seen through the entries of J, the “inverse” R(x ₁, x ₂) of J(f) = ∫₀ ¹ f”² .... is not to be per ceived intuitively. For the first question , there is a debate on whether closed form expressions are necessary versus efficient numerical algorithms. Besides the artistic interest one may have for such formulas, t he right choice is certainly dependent on the ultimate use of the kernel. Due to the diversity of spaces and norms, there are few systematic principles for the derivation of a kernel formula. Nevertheless, we also present the study of a number of interesting ad hoc constructions