ABSTRACT
Polynomial splines concern functions defined on a continuous interval. This is the most common situation in practice. Nevertheless, many applications require modeling functions defined on domains other than a continuous interval. For example, for spatial data with measurements on latitude and longitude, the domain of the function is the Euclidean space R
2. Specific spline models were developed for different applications. It is desirable to develop methodology and software on a general platform such that special cases are dealt with in a unified fashion. Reproducing Kernel Hilbert Space (RKHS) provides such a general platform. This section provides a very brief review of RKHS. Throughout this
book, important theoretical results are presented in italic without proofs. Details and proofs related to RKHS can be found in Aronszajn (1950), Wahba (1990), Gu (2002), and Berlinet and Thomas-Agnan (2004). A nonempty set E of elements f, g, h, . . . forms a linear space if there
are two operations: (1) addition: a mapping (f, g)→ f + g from E ×E into E; and (2) multiplication: a mapping (α, f)→ αf from R×E into E, such that for any α, β ∈ R, the following conditions are satisfied: (a) f + g = g+ f ; (b) (f + g)+ h = f +(g+ h); (c) for every f, g ∈ E, there exists h ∈ E such that f + h = g; (d) α(βf) = (αβ)f ; (e) (α + β)f = αf + βf ; (f) α(f + g) = αf + αg; and (g) 1f = f . Property (c) implies that there exists a zero element, denoted as 0, such that f + 0 = f for all f ∈ E. A finite collection of elements f1, . . . , fk in E is called linearly inde-
pendent if the relation α1f1+ · · ·+αkfk = 0 holds only in the trivial case with α1 = · · · = αk = 0. An arbitrary collection of elements A is called linearly independent if every finite subcollection is linearly independent. Let A be a subset of a linear space E. Define
spanA , {α1f1 + · · ·+ αkfk : f1, . . . , fk ∈ A, α1, . . . , αk ∈ R, k = 1, 2, . . . }.
