On orthogonalization of Schoenberg splines

The article is devoted to the application of the author's orthogonalization procedure of finite functions, which does not destroy their finite supports, to Schoenberg splines of the third degree. A general algorithm for modifying the Schoenberg mother spline within the framework of this orthogonalization procedure is described. It is shown that orthogonalization of the grid set of splines generated by the Schoenberg spline is achieved without changing the finite supports of the splines in the case of using eight step functions to modify the mother spline. Sixteen variants of orthogonalization for the cubic Schoenberg splines by step functions are found. In the first group of eight variants, all coefficients of the modifying step functions have real values, but the Schoenberg splines after such modification are not even or odd functions. In each of the eight variants of the second group, two coefficients are complex, and the remaining six coefficients have real values. The modified Schoenberg splines of the second group are sums of even and odd functions. A theorem on the order of approximation of any function from the Sobolev space by linear combinations of constructed orthogonal Schoenberg splines is proved.