Toeplitz matrices $A_{n}(f)$ is defined as:
$A_{n}(f)_{i,j}=c_{i-j}$ $0\leq i,j \leq n-1$, where $c_{k}$ are Fourier Coefficients of $f(\theta)=\sum_{k=- \infty}^{+\infty}c_{k}e^{\iota k \theta }$
$n$ here is the order of matrix $A_{n}(f)$.
Szegos Limit Theorems talk about determinants of large Toeplitz matrices. I am here concerned with the first Szegos limit theorem which states that: \begin{equation} lim_{n\rightarrow\infty}\frac{Det(A_{n}(f))}{Det(A_{n-1}(f))}=exp\{\frac{1}{2\pi}\int_{0}^{2\pi}log(f(\theta))d \theta\}\tag{1} \end{equation} Another form of Szegos First Limit Theorem is: \begin{equation} lim_{n\rightarrow\infty}(Det(A_{n}(f))^{\frac{1}{n}}=exp\{\frac{1}{2\pi}\int_{0}^{2\pi}log(f(\theta))d \theta\}\tag{2} \end{equation} I have example where it seems that Szegos lemma (1) is not true while (2) is true. This conclusion is based on plotting some initial values. Example:Take, \begin{equation} c_{n}= \begin{cases} e^{\iota\frac{\pi}{2}n} & n\,is\,even\\ \frac{1}{n^{2}+1}& n\,is\,odd\\ \end{cases} \ \end{equation} Here (1) seems to be untrue and (2) seems to be correct based on computation. From basic real analysis if (1) is true (2) has to be true, but is there any general criterion under which (2) is true and (1) is not?
Thank you