Abstract :
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.
