2.11 Let ????(h) = ????#(r), r = |h|, be an isotropic covariance function in 3, with spectral density f(????) = f #(????), ???? = |????|.

2.11 Let ????(h) = ????#(r), r = |h|, be an isotropic covariance function in ℝ3, with spectral density f(????) = f #(????), ???? = |????|. In addition to the usual integrability condition ∫ f(????) d???? < ∞, suppose also that ∫ |????|

2f(????) d???? < ∞.

(a) Show that ????(h) is twice continuously differentiable.

(b) Show that −????2????(h)∕????h[1]

2 is positive definite with spectral density

????[1]

2f(????).

(c) Hence, show that the negative Laplacian − △ ????(h)=− ∑3 j=1 ????2????(h)∕

????h[j]

2 is positive definite with isotropic radial spectral density ????2f #(????).

(d) Thus in d = 3 dimensions, deduce that the covariance function of the one-dimensional process needed in the turning bands algorithm in Section 2.14.3 is given by ????1(r)=−d2????#(r)∕dr2 − (2∕r)d????#(r)∕dr.