Twice continuously differentiable S-splines

Abstract

Twice continuously differentiable S-splines consisting of fifth degree polynomials are constructed, uniqueness and existence theorems are proved, stability conditions are established for such splines. The first three coefficients of each polynomial are determined by conditions of smooth gluing, the others are determined by the least squares method. This provides the ability to smooth initial data. The peculiarity of these splines is their semilocal property, i.e., each polynomial implicitly depends on function values determining previous polynomials and does not depend on values determining subsequent polynomials. It turns out that in this case the stability conditions are fulfilled under some very strong restrictions. Under there conditions and other ones ensuring sufficient closeness of the first polynomial and its derivatives to values of the function and its derivatives it is proved that this closeness is retained on the whole given interval.