Vallee Poussin theorem and Remez algorithm in the case of generalised degree polynomial spline approximation

The classical Remez algorithm was developed for constructing the best polynomial approximations for continuous and discrete functions in an interval. In this paper the classical Remez algorithm is generalised to the problem of polynomial spline (piece-wise polynomial) approximation with the spline defect equal to the spline degree. Also, the values of the splines in the end points of the approximation interval may be fixed. Polynomial splines combine simplicity of polynomials and flexibility, which allows one to significantly decrease the degree of the corresponding polynomials and oscillations of deviation functions. Therefore polynomial splines are a powerful tool for function and data approximation. The generalisation of the Remez algorithm developed in this research has been tested on several approximation problems. The results of the numerical experiments are presented