Hilbert schemes of points on some K3 surfaces and Gieseker stable bundles

By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces X the Hilbert schemes Hilb(n)(X) can be identified for all n greater than or equal to 1 with moduli spaces of Gieseker stable vector bundles on X. We also introduce a new Fourier-Mukai type transform for such surfaces.