The Space of Splines

Abstract

Because of its simple structure and good approximation properties, algebraic polynomials are widely used in practice for approximating of functions. The degree of this approximation depends essentially on the degree of the polynomial and the length of the considered interval [a, b]. Since the computation operations on polynomials of high degree involve certain problems it is advisable to use polynomials of low degree. In such a case, in order to achieve the desired accuracy we have to restrict ourselves to a small interval. For this purpose, one usually divides the original interval of consideration [a,b] into sufficiently small subintervals $\left\{ {\left[ {{x_k},{x_{k + 1}}} \right]} \right\}_{k = 0}^n$ and then uses a low degree polynomials p _k for approximation over $\left\{ {\left[ {{x_k},{x_{k + 1}}} \right]} \right\}$,..., n. This procedure produces a piecewise polynomial approximating function s(x),

$$ s\left( x \right) \equiv {p_k}\left( x \right)\quad on\quad \left\{ {\left[ {{x_k},{x_{k + 1}}} \right]} \right\},\quad k = 0, \ldots n $$

.