Mixed spline approximation

The mixed spline approximation problem combines the peculiarities of the problems of spline interpolation and smoothing that were studied by many mathematicians. The monograph of Loran should be mentioned specially. It gives the conditions of existence and uniqueness of the interpolating and smoothing splines in the general form. In practice the mixed problems were considered earlier (however, without the use of the variational formulation). The main purpose of the given paper is to obtain precise conditions of unique solvability of the mixed problem.