G. Szegö's limit theorem in the multidimensional case (Q796807)

Let \(T^ n\) be an n-dimensional torus and let \(\Phi\) (t) be an integrable function on \(T^ n\) with Fourier coefficients \(\phi\) (\(\alpha)\), \(\alpha \in Z^ n\). Let also \(F_{\Phi}f=\sum_{\beta \in Z^ n}\phi(\alpha -\beta)f(\beta)\) be a convolution operator. The author considers the operator \(P_{\tau M}F_{\Phi}P_{\tau M}\) where \(P_ M\) is a projector in \(\ell_{\infty}(Z^ n)\) acting in the form: \(P_ M(f)(\alpha)=f(\alpha)\) for \(\alpha \in M\), \(P_ M(f)(\alpha)=0\) for \(\alpha \not\in M\). The behaviour of the determinant \(D_{\tau M}[\Phi]\) of the operator \(P_{\tau M}F_{\Phi}P_{\tau M}\) when \(\tau \to \infty\) is described in the one-dimensional case by Szegö's theorem. The generalization of the Szegö's theorem to the case \(n>1\) was given by I. Yu. Linnik. The author suggests certain modifications of Linnik's result.