On interpolation by almost trigonometric splines

The existence and uniqueness of an interpolating periodic spline defined on an equidistant mesh by the linear differential operator $\mathcal{L}_{2n+2}(D)=D^{2}(D^{2}+1^{2})(D^{2}+2^{2})\cdots (D^{2}+n^{2})$ with $n\in\mathbb{N}$ are reproved under the final restriction on the step of the mesh. Under the same restriction, sharp estimates of the error of approximation by such interpolating periodic splines are obtained.