Prove the following generalization of Lemma 17.1. Figure 17.5. Decomposition of the kernel of bounded variation into the difference of monotonically decreasing kernels. Suppose K : RR is bounded...

Prove the following generalization of Lemma 17.1.

Figure 17.5. Decomposition of the kernel of bounded variation into the difference of monotonically decreasing kernels.

Suppose K : R→R is bounded and

K(||x||) ∈ L1(λ) ∩ Lp(λ)

for some p ∈ [1, ∞) and assume that ‑ K(||x||) dx = 0. Let µ be an arbitrary probability measure on Rd and let q ∈ (0, ∞). Then the RBF networks in the form (17.8) are dense in both Lq(µ) and Lp(λ). In particular, if m ∈ Lq(µ)∩Lp(λ), then for any there exists a θ = (w0,...,wk, b1,...,bk, c1,...,ck) such that

Rd |fθ(x) − m(x)| q µ(dx) <>