Title

An N-dimensional analogue of Szegö's limit theorem

Document Type

Article

Publication Title

Journal of Mathematical Analysis and Applications

Publication Date

2-15-1996

Abstract

Let T denote the unit circle in the complex plane, let φ ∈ L∞(TN), and for (a1,...,aN) ∈ (ℤ+\0)N let Λ be the rectangular lattice: Λ = {(m1,...,mN) ∈ ℤN: mi ∈ [0,ai)}. For each positive integer p define Λp = {(m1,...,mN) ∈ ℤN: mi ∈ [0,pai)}. The Toeplitz matrix Τp(φ): l2(Λp) → l2(Λp) with symbol φ is defined by (equation presented) where {φ̂(m)}m∈ℤN denotes the Fourier coefficients of φ. Assuming appropriate conditions on φ and on a function F we find N + 1 terms of the asymptotic expansion of the trace of F(Τp(φ)) as p → ∞. This expansion takes the form (equation presented) where the coefficients cJ,F = cJ,F (φ) depend on the N - J dimensional 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.

Volume

198

Issue

1

First Page

137

Last Page

165

DOI

10.1006/jmaa.1996.0073

ISSN

0022247X

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