Scientific publications

If a function ƒ has a sectionally continuous derivative of order п bounded in sections of continuity, then it can be smoothed up to the function having the continuous derivative of the order no less than п. The smoothing can be done by summing with the algebraic spline of degree п + 1 with defect 1, which is determined in an arbitrarily small one-sided neighborhood of the node, where there is the gap of the п-th derivative of f. In addition, it is possible to save values of lower order derivatives in the node and disrupt the original upper estimation of the п-th derivative module in the whole domain of its definition only up to an arbitrarily small value. If f additionally has continuous derivatives f^{(n + 1)}, …, f^{(n + k)} in the domain С of continuity of f⁽ⁿ⁾, then the smoothing with the algebraic spline S of degree n + к + 1, in addition to above mentioned properties can ensure continuity of the sum of derivatives (f+S)⁽ⁿ⁺¹⁾, n = 1,...,к in the domain С.