Mirror symmetry on K3 surfaces via Fourier-Mukai transform

We use a relative Fourier-Mukai transform on elliptic K3 surfaces X to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of X reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on X which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry. From the mathematical viewpoint the novelty is that we exhibit another example of a Fourier-Mukai transform on K3 surfaces, whose properties are closely related to the geometry of the relative Jacobian of X.