  Let $q\ge 1$ be an integer. Given $M$ samples of a smooth function of $q$
variables, $2\pi$--periodic in each variable, we consider the problem of
constructing a $q$--variate trigonometric polynomial of spherical degree
$\O(M^{1/q})$ which interpolates the given data, remains bounded (independent
of $M$) on $[-\pi,\pi]^q$, and converges to the function at an optimal rate on
the set where the data becomes dense. We prove that the solution of an
appropriate optimization problem leads to such an interpolant. Numerical
examples are given to demonstrate that this procedure overcomes the Runge
phenomenon when interpolation at equidistant nodes on $[-1,1]$ is constructed,
and also provides a respectable approximation for bivariate grid data, which
does not become dense on the whole domain.
