Abstract
The emphasis on polynomials is justified by Weierstrass's theorem, which assures that by choosing a polynomial of sufficiently high degree one can approximate closely to a given function. However, there are cases where the convergence of polynomial approximants is very slow and this leads to the development of other types of approximations. For a given partition of [a, b], the spline consists of n polynomial segments with the appropriate continuity condition at each of the interior knots, which gives the spline a certain degree of smoothness. It is rare to use a spline of higher order than the cubic spline, which is the one most commonly used. For the general spline of degree k, the k + 1 conditions are required to determine the polynomial of degree k on each interval. Thus, to determine a spline of degree k, one requires n( k + 1) – (n – 1)k = n + k conditions.
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