Smoothness of Generalized Eigenfunctions of Differential–Difference Operators on a Finite Interval

The eigenfunction–eigenvalue problem for differential–difference operators is considered. Necessary and sufficient conditions for preserving the smoothness of generalized eigenfunctions over the entire interval are obtained. An example is given of a differential–difference operator having a countable set of eigenfunctions whose smoothness is violated inside the interval and a countable set of eigenfunctions whose smoothness is preserved.