De Branges - Rovnyak spaces of discrete analytic functions on a rhombic lattice

There are several known generalizations of the Beurling - Lax theorem that establish a correspondence between shift - invariant spaces of analytic functions and Schur functions. The situation becomes more complicated when the usual notion of analyticity is replaced with a discrete analogue due to J. Ferrand and R. J. Duffin. It turns out that for a suitably defined shift operator, one can still describe shift - invariant reproducing kernel Hilbert spaces of discrete analytic functions on a rhombic lattice in terms of the convolution product and discrete analytic Schur functions. This is joint work with D. Alpay and A. Fuerte Perez.