  We study a minimum problem and associated maximum problem for finite,
complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size
$(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on $P_N$,
the space of complex polynomials of degrees at most $N$, with the usual Hilbert
space structure it inherits as a subspace of $L^2$ of the unit circle. The
operator $A$ is the compression to $P_N$ of the multiplication operator on
$L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of
indices between $-N$ and $N$ match the matrix entries of $A$. Our minimum
problem is to minimize the $L^{\infty}$ norm of such inducers. We show there is
a unique one of minimum norm, and we describe it. The associated maximum
problem asks for the maximum of the ratio of the preceding minimum to the
operator norm. That problem remains largely open. We present some suggestive
numerical evidence.
