The first Szegő limit theorem on multi-dimensional torus.

In this paper, we consider the first Szegő limit theorems on d -torus T d for 1 ≤ d ≤ + ∞. Given φ ∈ L 1 (T d) and a finite subset σ = { ξ 1 ⋯ , ξ n } of the dual group Z d of T d , the truncated Toeplitz matrix with respect to σ is T σ φ = { φ ˆ (ξ j − ξ i) } 1 ≤ i , j ≤ n. For any Følner sequence { σ N } of Z d and φ ∈ L + 1 (T d) , it is shown that lim N → ∞ ⁡ (det ⁡ T σ N φ) 1 | σ N | = exp ⁡ (∫ T d log ⁡ φ d m d). In the case d = + ∞ , we are associated with multiplicative Toeplitz matrix T φ = { φ ˆ (j / i) } i , j ∈ N and the most relevant non-Følner truncation, that is, T N φ = { φ ˆ (j / i) } 1 ≤ i , j ≤ N , where σ N = { 1 , ... , N }. It is shown that for each φ ∈ L R ∞ (T ∞) and f ∈ C [ ess-inf φ , ess-sup φ ] , the limit lim N → ∞ ⁡ 1 N Tr f ( T N φ) exists. Moreover, it is proven that the limit lim N → ∞ ⁡ (det ⁡ T N φ) 1 N exists for any φ ∈ L + 1 (T ∞) with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski in [30]. [ABSTRACT FROM AUTHOR]