The Problem of Topologies of Grothendieck and the Class of Fréchet T-Spaces

We examine the interpolation with periodic polynomial splines of degree d and defect r (d ≦ r) on equidistant partitions of the real axis and generalize known results for r = 0. We prove necessary and sufficient conditions for the existence and a certain L²-stability of the interpolants as well as their approximation properties in the scale of the periodic SOBOLEV spaces.