Theory

Abstract

A Reproducing Kernel Hilbert Space (RKHS) is first of all a Hilbert space, that is, the most natural extension of the mathematical model for the actual space where everyday life takes place (the Euclidean space ℝ3). When studying elements of some abstract set S it is convenient to consider them as elements of some other set S′ on which is already defined a structure relevant to the problem to be treated. It can be for instance an order structure, a vector structure, a metric structure or a mixing of algebraic and topological structures. For this we need an “imbedding theorem” or a “representation theorem”. Through this kind of theorem the study of elements of S is transferred to their “representers” in S′ and can be carried out using the structure on S′. For their richness and simplicity Hilbert spaces are introduced as often as possible when a vector structure and an inner product can be exploited. They provide powerful mathematical tools and geometric concepts on which our intuition can rest. The phrase “RKHS method” is generic to name a method based on the embedding of the abstract set S into some RKHS S′.