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 $1\le d\le +\infty$. It is shown that for any Følner sequence $\{{\sigma }_N\}$ of ${\mathbb{Z}}^d$ and $\phi \in L_{+}^1({\mathbb{T}}^d)$, it holds that $$\underset{N\to \infty }{\lim }{\left(\det T_{{\sigma }_N}\phi \right)}^{\frac{1}{|{\sigma }_N|}}=\exp \left({\int }_{{\mathbb{T}}^d}\log \phi d{m}_d\right).$$ In the case $d=+\infty$, we are associated with multiplicative Toeplitz matrix $T\phi =\{\hat{\phi }(j/i){\}}_{i,j\in \mathbb{N}}$ and the most concerned non-Følner truncation, that is, $T_N\phi =\{\hat{\phi }(j/i){\}}_{1\le i,j\le N}$, where ${\sigma }_N=\{1,\dots ,N\}$. It is shown that for each $\phi \in L_{\mathbb{R}}^{\infty }({\mathbb{T}}^{\infty })$ and $f\in C[\text{ess-inf}\phi ,\text{ess-sup}\phi ]$, the limit ${\lim }_{N\to \infty }\frac{1}{N}\mathrm{Tr}f\left(T_N\phi \right)$ exsits. Moreover, it is proven that the limit ${\lim }_{N\to \infty }{\left(\det T_N\phi \right)}^{\frac{1}{N}}$ exists for any $\phi \in L_{+}^1({\mathbb{T}}^{\infty })$ with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski.

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