We investigate the multivariate sampling theory associated with multiparameter eigenvalue problems. A several-variable counterpart of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon is given. It arose when the multiparameter system has order one. Two-dimensional sampling theorems associated with two-parameter systems of second-order differential operators will be established. The sampling formulae are of multivariate non-uniform Lagrange interpolation type. Unlike many of the known formulae, the interpolating functions are not necessarily products of single variable functions.
