A Fourier-Mukai transform for stable bundles on K3 surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface X is here played by a suitable component X^ of the moduli space of stable sheaves on X. For a wide class of K3 surfaces X^ can be chosen to be isomorphic to X; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle F is stable and has the same Euler characteristic as F