TangentKernels

The kernel Θ(L)(x,z) is defined using the following recursive definition. When β = 0, the recursive formulation is the same as existing derivations, e.g., [9]. We split the theorem into the next two lemmas. Lemma 2. Let x,z Sd 1 and kFCβ(2)(xTz) as defined in(5) with β > 0. This completes the proof, by using Aronszan's inclusion theorem as follows. The first integral, A(k,d,c), was shown in(14) to convergeasymptotically toB2k d. (c 2) Denote by k0 the smallest k for which Theorems 1 and 2 hold simultaneously.