AnN-Dimensional Analogue of Szegö's Limit Theorem

AbstractLet T denote the unit circle in the complex plane, let φ∈L∞(TN), and for (a1,…,aN)∈(Z+\0)Nlet Λ be the rectangular lattice: Λ={(m1,…,mN)∈ZN:mi∈[0,ai)}. For each positive integerpdefine Λp={(m1,…mN)∈ZN:mi∈[0,pai)}. The Toeplitz matrixTp(φ):l2(Λp)→l2(Λp) with symbol φ is defined by[formula]where[formula]denotes the Fourier coefficients of φ. Assuming appropriate conditions on φ and on a functionFwe findN+1 terms of the asymptotic expansion of the trace ofF(Tp(φ)) asp→∞. This expansion takes the form[formula]where the coefficientscJ,F=cJ,F(φ) depend on theN−Jdimensional faces of Λ. We also find an expansion for the case when the edges of Λ expand at different rates and we apply this generalization to compute an example