On Kotelnikov-Shannon theorem for random fields on a cylinder (Q2755280)

The paper deals with the mean square continuous random fields \(\xi(t,\varphi)\), \(t\in{\mathbf R}\), \(\varphi\in[-pi,+pi]\) homogeneous with respect to time isotropic on the unit sphere \(S_2\). This type of fields has been considered in a number of papers [see, for example, \textit{M. P. Moklyachuk} and \textit{M. I. Yadrenko}, Theory Probab. Math. Stat. 18, 115-124 (1979); translation from Teor. Veroyatn. Mat. Stat. 18, 106-115 (1978; Zbl 0409.62087) and ibid. 19, 129-139 (1980); resp. ibid. 19, 111-121 (1978; Zbl 0407.60059)]. To approximate the fields the authors use the fields NEWLINE\[NEWLINE\xi_N(t,\varphi)= \sum_{k=-n}^n\xi\left({{l\pi}\over{c}},\varphi\right) {{\sin(c(t-l\pi/c))}\over {c(t-l\pi /c)}}.NEWLINE\]NEWLINE The mean square error \(E|\xi(t,\varphi)-\xi_N(t,\varphi)|^2\) of the approximation is estimated. It is proved that for random fields with bounded spectrum the Kotelnikov-Shannon formula (sampling theorem) NEWLINE\[NEWLINE\xi(t,\varphi)= \sum_{k=-\infty}^{+\infty} \xi\left({{l\pi}\over{c}},\varphi\right) {{\sin(c(t-l\pi/c))}\over {c(t-l\pi/c)}}NEWLINE\]NEWLINE holds true.