The first Szegő limit theorem on multi-dimensional torus

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