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