2.5 (Matrn, 1960, p. 16) Using Eq. (2.26) show that an isotropic radial covariance function ????#(r), r

2.5 (Matérn, 1960, p. 16) Using Eq. (2.26) show that an isotropic radial covariance function ????#(r), r ≥ 0, in d dimensions is bounded below by inf r≥0

{????#(r)}∕????#(0) ≥ Γ(???? + 1)inf x>0

{(x∕2)

−????J????(x)} = Ld,say, where ???? = 1 2

(d − 2). Show that the shell scheme, for which the radial spectral measure F#(d????) is a point mass (item 9 in Section 2.7), attains this lower bound. The first few values of this lower bound are L1 = −1, L2 = −0.403, L3 = −0.217, L4 = −0.132; and limd→∞Ld = 0. Thus, only limited amounts of negative autocorrelation are allowed in isotropic random fields in ℝd, for d > 1.