Two-sided semi-local smoothing splines

A semi-local smoothing spline of degree $n$ and class $C^p$ is a function defined on an interval, having $p$ continuous derivatives on that interval, and coinciding with a polynomial of degree $n$ on the subintervals forming its partition. The domain of each polynomial is a subinterval on which $m+1$ values of the approximated function are given, but in order to construct the polynomial, it is necessary to know $M\geqslant m+1$ values ($m$ and $M$ are determined by the class and the degree of the spline). The additional values can be borrowed from the adjacent subintervals. When constructing an $S$-spline in the periodic case, the problem of additional values is solved on the basis of periodicity, but in the nonperiodic case, one is expected to define the lacking values of a function beyond the domain. The present paper is aimed at nonperiodic two-sided $S$-splines whose construction does not require additional data.