  We investigate the weighted Bojanov-Chebyshev extremal problem for
trigonometric polynomials, that is, the minimax problem of minimizing
$\|T\|_{w,C({\mathbb T})}$, where $w$ is a sufficiently nonvanishing, upper
bounded, nonnegative weight function, the norm is the corresponding weighted
maximum norm on the torus ${\mathbb T}$, and $T$ is a trigonometric polynomial
with prescribed multiplicities $\nu_1,\ldots,\nu_n$ of root factors
$|\sin(\pi(t-z_j))|^{\nu_j}$. If the $\nu_j$ are natural numbers and their sum
is even, then $T$ is indeed a trigonometric polynomial and the case when all
the $\nu_j$ are 1 covers the Chebyshev extremal problem.
  Our result will be more general, allowing, in particular, so-called
generalized trigonometric polynomials. To reach our goal, we invoke Fenton's
sum of translates method. However, altering from the earlier described cases
without weight or on the interval, here we find different situations, and can
state less about the solutions.
