An N-Dimensional Analogue of Szeg¨o’s Limit Theorem

Let T denote the unit circle in the complex plane, let f gL` TN ., and for a1, . . . , aN.g Zq_0.N let L be the rectangular lattice: Ls m1, . . . , mN .gZN: migw0, ai.4. For each positive integer p define Lps m1, . . . , mN.gZN: mig w0, pai.4. The Toeplitz matrix Tp f.: l 2 Lp.ªl2 Lp.with symbol f is defined by Tp f.a m.s fˆ myn.a n. ngLp where fˆ m.4mgZN denotes the Fourier coefficients of f. Assuming appropriate conditions on f and on a function F we find Nq1 terms of the asymptotic expansion of the trace of F Tp f..as pª`. This expansion takes the form N trF Tp f..s cJ, FPNyJqo 1., js0 where the coefficients cJ,FscJ,F f. depend on the NyJ dimensional faces of L. We also find an expansion for the case when the edges of L expand at different rates and we apply this generalization to compute an example.