  It is shown, that any sufficiently smooth periodic solution of the
self-focusing Nonlinear Schr\"odinger equation can be approximated by periodic
finite-gap ones with an arbitrary small error. As a corollary an analogous
result for the motion of closed curves in ${\Bbb R}^3$ guided by the Filament
equation is proved. This equation describes the dynamics of very thin filament
vortices in a fluid.
