The first Szegő limit theorem on multi-dimensional torus

Abstract

In this paper, we consider the first Szegő limit theorems on $d$-torus $\mathbb{T}^d$ for $1\leq d\leq +\infty$. It is shown that for any Følner sequence $\{σ_N\}$ of $\mathbb{Z}^d$ and $φ\in L^1_+(\mathbb{T}^d)$, it holds that $$ \lim_{N\rightarrow \infty}\left(\det T_{σ_N}φ\right)^{\frac{1}{|σ_N|}}=\exp\left(\int_{\mathbb{T}^d} \logφ~dm_{d}\right). $$ In the case $d=+\infty$, we are associated with multiplicative Toeplitz matrix $T φ=\{\widehatφ(j/i)\}_{i,j\in\mathbb{N}}$ and the most concerned non-Følner truncation, that is, $T_N φ=\{\widehatφ(j/i)\}_{1\leq i,j\leq N}$, where $σ_N=\{1,\dots,N\}$. It is shown that for each $φ\in L^\infty_{\mathbb{R}}(\mathbb{T^{\infty}})$ and $f\in C[\text{ess-inf} ~φ,~\text{ess-sup}~φ]$, the limit $\lim_{N\rightarrow \infty} \frac{1}{N}\mathrm{Tr} f \big(T_N φ\big)$ exsits. Moreover, it is proven that the limit $\lim_{N\rightarrow \infty}\left(\det T_N φ\right)^{\frac{1}{N}}$ exists for any $φ\in L^1_+(\mathbb{T}^\infty)$ with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski.