Abstract
In this chapter, we identify extensions of the initially give positive definite (p.d.) functions F which are associated with operator extensions in the RKHS \(\mathcal{H}_{F}\) itself (Type I), and those which require an enlargement of \(\mathcal{H}_{F}\), Type II. In the case of \(G = \mathbb{R}\) (the real line) some of these continuous p.d. extensions arising from the second construction involve a spline-procedure, and a theorem of G. Pólya, which leads to p.d. extensions of F that are symmetric around x = 0, and convex on the left and right half-lines. Further these extensions are supported in a compact interval, symmetric around x = 0.
“Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either.”—Ian Stewart, Preface to second edition of What is Mathematics?
by Richard Courant and Herbert Robbins
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Jorgensen, P., Pedersen, S., Tian, F. (2016). Type I vs. Type II Extensions. In: Extensions of Positive Definite Functions. Lecture Notes in Mathematics, vol 2160. Springer, Cham. https://doi.org/10.1007/978-3-319-39780-1_5
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