On some linear operations of local averaging of homogeneous random fields (Q2767617)

Let \(\xi(t_1,t_2)\) be a homogeneous random field with zero mean. The author finds spectral densities and joint spectral densities of the homogeneous random fields NEWLINE\[NEWLINE\alpha_{T_1,T_2}(t_1,t_2)={1\over 2(T_1+T_2)}\left\{\int_{t_1-T_1}^{t_1+T_1}\xi(u,t_2) du+ \int_{t_2-T_2}^{t_2+T_2}\xi(t_1,v) dv\right\}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \beta_{T_1,T_2}(t_1,t_2)={1\over 2}\left\{{1\over 2T_1}\int_{t_1-T_1}^{t_1+T_1}\xi(u,t_2) du+ {1\over 2T_2}\int_{t_2-T_2}^{t_2+T_2}\xi(t_1,v) dv\right\}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE l_{T}(t_1,t_2)={1\over 2T^2}\iint_{|u-t_1|+|v-t_2|\leq T}\xi(u,v) du dv,\quad\psi_{T,D}(t_1,t_2)={1\over T^2|D|}\iint_{S_{t_1,t_2}}\xi(u,v) du dv, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \gamma_{T_1,T_2}(t_1,t_2)=\tfrac 14[\xi(t_1-T_1,t_2)+\xi(t_1+T_1,t_2)+ \xi(t_1,t_2+T_2)+\xi(t_1,t_2-T_2)],NEWLINE\]NEWLINE where \( S_{t_1,t_2}=\{(u,v):T^{-1}(t_1-u,t_2-v)\in D\},\;D\subset R^2\). Some applications of the obtained results to the estimation of the unknown mean of a homogeneous random field are presented.