Interval Estimation using Integro-Differential Splines of the Third Order of Approximation

Interpolation by local splines in some cases gives a better result than other splines or interpolation by classical interpolation polynomials. Integro-differential splines are one of the types of local splines that use, in addition to the values of the functions at the nodes of the grid, integrals over grid intervals. To construct an approximation on a finite interval, in order to improve the approximation quality, we use left or right integrodifferential splines near the ends of this interval. At some distance from the ends, besides the left or right splines, we can also use the middle integro-differential splines. Sometimes it is not necessary to calculate the approximation of the function at intermediate points in [𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑗𝑗+1]. Instead of calculating the approximation in many points in [𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑗𝑗+1] it is sufficient to estimate only the upper and lower boundaries of the variety of the approximation on this interval. The paper discusses the estimation of the boundaries of approximation of functions using left, right and middle trigonometrical integro-differential splines of the third order of approximation. The process of constructing the basic splines is discussed. The approximation theorems are given. Unimprovable constants in the approximation inequalities are given. Numerical examples of construction of the approximations and interval estimation are given.