Generalization of G. Szegö's limit theorem to the multidimensional case
Let \(T^ n=\{t=(t_ 1,...,t_ n)\in C^ n:| t_ i| =1\), \(1\leq i\leq n\}\) be the n-dimensional torus and dt be the Lebesgue measure on it such that \(\int_{T^ n}dt=1\); and \(Z^ n=\{\alpha =(\alpha_ 1,...,\alpha_ n)| \alpha_ i\in Z\), \(1\leq i\leq n\}\) be the integral lattice in \(R^ n\). For \(t\in T^ n\) and \(\alpha \in Z^ n\), let us set \(t^{\alpha}=t_ 1^{\alpha_ 1}...t_ n^{\alpha_ n}\), \(| \alpha | =| \alpha_ 1| +...+| \alpha_ n|.\) Let \(\Phi\) be a function, defined and integrable on \(T^ n\), and \(\phi (\alpha)=\int_{T^ n}t^{-\alpha}\Phi (t)dt\) \((\alpha \in Z^ n)\) be its Fourier coefficients. Let us denote the Fourier coefficients of ln \(\Phi\) by s(\(\alpha)\). For a bounded set \(M\subset R^ n\), let us set \(\tau M=\{x\in R^ n| x/\tau \in M;r=1,2,...\};\) and denote the number of elements of the lattice \(Z^ n\) contained in M by \(M^{\#}\). We introduce the following two operators in the space of functions on \(Z^ n:\) \((F_{\Phi}f)(\alpha)=(\phi *f)(\alpha)=\sum_{\beta \in Z^ n}\phi (\alpha -\beta)f(\beta)\), \((P_ Mf)(\alpha)=f(\alpha)\), \(\alpha \ni M\cap Z^ n=0\), \(\alpha\) \(\not\in M\). In the present article, we study the behavior of the determinant \(D_{\tau M}[\Phi]\) of the operator \(P_{\tau M}F_{\Phi}P_{\tau M}\) for \(\tau\) \(\to \infty\) (\(\tau\in N)\).
scientific article; zbMATH DE number 3912952
scientific article