Twice continuously differentiable semilocal smoothing spline

Twice continuously differentiable periodic local and semilocal smoothing splines, or $S$-splines from the class $C^2$ are considered. These splines consist of polynomials of 5th degree, first three coefficients of each polynomial are determined by values of the previous polynomial and two its derivatives at the point of splice, coefficients at higher terms of the polynomial are determined by the least squares method. These conditions are supplemented by the periodicity condition for the spline function on the whole segment of definition or by initial conditions. Uniqueness and existence theorems are proved. Stability and convergence conditions for these splines are established.