Numerical construction of the Hill functions

In approximation theory or for the numerical solution of ordinary or partial differential equations, functions are often approximated by finite sums of the form \[f(x) = \sum _k {c_k \varphi \left( {\frac{{x - kh}}{h}} \right),} \] where $\varphi (x)$ must satisfy certain conditions. Frequently these functions are given by a convolution formula\[\varphi_n (x) = \varphi _{n - 1} (x) * \varphi _1 (x),\] where $\varphi _1 (x) = 1$ for $x \in \langle - \frac{1}{2},\frac{1}{2} \rangle ,\varphi _1 (x) = 0$ otherwise. An algorithm for computing $\varphi _n (x)$ for higher values of n is needed because these functions are useful in solving differential equations of higher order.The paper concerns the numerical construction of these functions as well as their derivatives. An algorithm with reasonable stability is designed using local coordinates and expansion in terms of Legendre polynomials. The orthogonality of Legendre polynomials makes the computations involved in approximation of the above form very simple wh...