Radial kernels and their reproducing kernel Hilbert spaces

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Consistent Kernel Mean Estimation for Functions of Random Variables (2016, background): “Using Scovel et al. (2014, Corollary 4.9.i) together with Steinwart and Christmann (2008, Theorem 4.26.i) we conclude that T θ/2 is injective. Thus μi > 0 for all i. Thus, if θ = 1, Lemma 6.4 of Steinwart and Scovel (2012) shows that the range of T θ/2 is [Hk]∼, the image of the canonical embedding ”
Robust kernel density estimation (2008, background): “ne, 4 i.e., ( 1)k dk dtk g(t) 0; for all k1;t>0; lim t!0 g(t) = g(0); and to the assumption that there exists a nite Borel measure on R+ , [0;1) such that k ˙(x;x0) = Z exp t2kx x0k2 d(t): See Scovel et al. (2010). Well-known examples of kernels satisfying all of the above properties are the Gaus”
Universal Kernels on Non-Standard Input Spaces (2010, background): “Indeed, [25] uses the fact that on R such kernels have an integral representation in terms of the Gaussian RBF kernels to show, see [25, Corollary 4.”