On orthogonal cubic Schoenberg splines

The modification of the mother cubic Schoenberg spline is carried out using four cubic Schoenberg splines having finite supports, the sizes of which are smaller compared to the size of the finite support of the mother spline. As a result, eight grid sets of orthogonal cubic Schoenberg splines with real values are constructed. A theorem on the order of approximation of any function of the Sobolev space by linear combinations of constructed orthogonal cubic Schoenberg splines is proved. It is shown that the order of approximation by Schoenberg splines, also modified by Schoenberg splines, is significantly higher than the order of approximation by Schoenberg splines modified by step functions, and coincides with the order of approximation by classical cubic Schoenberg splines. The defect of the modified Schoenberg spline is equal to one, as that of the classical Schoenberg spline. A modified spline is a continuous function in which there are no breaks in the first and second derivatives at the points where the parts of the mother spline and the parts of the splines used for modification meet.