Find formulas in terms of $\gamma_{i j}^{(k)}$ analogous to those given in the preceding exercise for the factors in a triangular factorization of the form
$$
\Gamma_n=U_n D_n U_n^H,
$$
where $T_n \succ O, U_n$ is an upper triangular matrix and $D_n$ is a diagonal matrix.
Positive definite Toeplitz matrices play a significant role in the theory of prediction of stationary stochastic sequences, which, when recast in the language of trigonometric approximation, focuses on evaluations of the following sort:
$$
\min \left\{\frac{1}{2 \pi} \int_0^{2 \pi}\left|e^{i n \theta}-\sum_{j=0}^{n-1} c_j e^{i j \theta}\right|^2 f\left(e^{i \theta}\right) d \theta: c_0, \ldots, c_{n-1} \in \mathbb{C}\right\}=\left\{\gamma_{n n}^{(n)}\right\}^{-1}
$$
and
(12.23)
$$
\min \left\{\frac{1}{2 \pi} \int_0^{2 \pi}\left|1-\sum_{j=1}^n c_j e^{i j \theta}\right|^2 f\left(e^{i \theta}\right) d \theta: c_1, \ldots, c_n \in \mathbb{C}\right\}=\left\{\gamma_{00}^{(n)}\right\}^{-1},
$$
where, for ease of exposition, we assume that $f\left(e^{i \theta}\right)$ is a continuous function of $\theta$ on the interval $0 \leq \theta \leq 2 \pi$ such that $f\left(e^{i \theta}\right)>0$ on this interval. Let $T_n=T_n(f)$ denote the Toeplitz matrix with entries
$$
t_j=\frac{1}{2 \pi} \int_0^{2 \pi} f\left(e^{i \theta}\right) e^{-i j \theta} d \theta \quad \text { for } \quad j=0, \pm 1, \pm 2, \ldots .
$$