On proximity of correlation functions of homogeneous and isotropic random fields whose spectral functions coincide on a certain set (Q2739679)

Let \(\gamma_1(x),\gamma_2(x),x\in \mathbb R^n\), be square mean continuous homogeneous and isotropic random fields, let \(B_{n,1}(t),B_{n,2}(t)\) be the correlation functions of the fields, and let \(\Phi_{n,1}(\lambda),\Phi_{n,2}(\lambda)\) be the spectral functions of the fields. The author gives examples of application of the mean-value theorem to estimate the proximity of the correlation functions in the case where their spectral functions coincide on a certain set. If, for example, \(\Phi_{n,1}(\lambda)=\Phi_{n,2}(\lambda)\) for all \(\lambda\geq c\), then NEWLINE\[NEWLINE|B_{n,1}(t)-B_{n,2}(t)|\leq \left|2^{(n-2)/2} \Gamma\left({n}\over{2}\right) {{J_{(n-2)/2}(ct)}\over{(ct)^{(n-2)/2}}}-1\right|\sup_{x\in R}|\Phi_{n,1}(x)-\Phi_{n,2}(x)|NEWLINE\]NEWLINE for all \(t\in [0,1]\).