Inverse Spectral Problem for the Laguerre Differential Operator

We construct a self-adjoint operator associated with the following given set of eigenvalues: {an+c}_n≤-1∪{an+b}_n≥0. This problem arises in sampling theory and deals with the construction of an interpolating basis associated with a given set of sampling points, the so-called two sided cardinal series. In order to solve this inverse spectral problem we shall need to extend the Gelfand-Levitan theory to operators close to the Laguerre differential operator.