Semilocal smoothihg $S$-splines

Semilocal smoothing splines or $S$-splines from class $C^p$ are considered. These splines consistof polynomials of a degree $n$, first $p + 1$ coefficients of each polynomial are determined by values of the previous polynomial and $p$ its derivatives at the point of splice, coefficients at higher terms of the polynomial aredetermined by the least squares method. These conditions are supplemented by the periodicity condition for thespline 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.