Polynomial covariance functions on intervals (Q1395937)

Let \(C(t)=r-(1/2)|t|+a_2t^2+(\varepsilon a_3^2/12) |t|^3 + (a_4/24)t^4\) for \(|t|\leq 1\), where \(\varepsilon = \pm 1\). The authors describe conditions under which \(C(t)\) is a positive definite function and thus it is a covariance function. Their method is based on Krein-Langer theory on some operators in the space \(L^2[0,1]\). Special cases when \(a_4=0\) or \(a_3=0\) are analyzed in detail. The results are extended to isotropic random fields with applications in spatial statistics.