  Any sequence of uniformly bounded $N\times N$ Hermitian Toeplitz matrices
$\{\boldsymbol{H}_N\}$ is asymptotically equivalent to a certain sequence of
$N\times N$ circulant matrices $\{\boldsymbol{C}_N\}$ derived from the Toeplitz
matrices in the sense that $\left\| \boldsymbol{H}_N - \boldsymbol{C}_N
\right\|_F = o(\sqrt{N})$ as $N\rightarrow \infty$. This implies that certain
collective behaviors of the eigenvalues of each Toeplitz matrix are reflected
in those of the corresponding circulant matrix and supports the utilization of
the computationally efficient fast Fourier transform (instead of the
Karhunen-Lo\`{e}ve transform) in applications like coding and filtering. In
this paper, we study the asymptotic performance of the individual eigenvalue
estimates. We show that the asymptotic equivalence of the circulant and
Toeplitz matrices implies the individual asymptotic convergence of the
eigenvalues for certain types of Toeplitz matrices. We also show that these
estimates asymptotically approximate the largest and smallest eigenvalues for
more general classes of Toeplitz matrices.
