The mth difference operator applied to L-2 functions on a finite interval

Continuous and discrete versions of difference operators are introduced, involving special examples of Toeplitz matrices. A spectral analysis of the continuous operator is performed, providing a full characterization of the null space and the singular system. Under a simplifying assumption involving the discretization parameter, the same analysis is performed for the discrete operator, by showing that the singular value decomposition is preserved in some sense. Many correspondences are lost in the case where the simplifying assumption is not verified.