The paper is aimed at periodic and nonperiodic semilocal smoothing splines, or S-splines of class C_p, formed by polynomials of degree n. The first p + 1 coefficients of each polynomial are determined by the values of the preceding polynomial and its first p derivatives at the glue-points, while the remaining n-p coefficients of the higher derivatives of the polynomial are found by the method of least squares. These conditions are supplemented with the initial conditions (nonperiodic case) or the periodicity condition on the spline-function on the segment where it is defined. A linear system of equations is obtained for the coefficients of the polynomials constituting the spline. Its matrix has a block structure. Existence and uniqueness theorems are proved and it is shown that that the convergence of the splines to the original function depends on the eigenvalues of the stability matrix. Examples of stable S-splines are given.
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