Convergence of variational splines I

Strong convergence of interpolating splines on the imbedded meshes is established without the assumption that the system of operators corresponding to the added interpolation conditions is correct. It is also shown that correctness of the system of operators is equivalent to the zero intersection of their kernels.

The necessary and sufficient conditions of convergence of the mixed splines on the subspaces to the mixed spline on the whole space are obtained, and simple sufficient conditions of their convergence are found.