A second look at the interpolatory background of the Euler-Maclaurin quadrature formula

In [4] the first named author discussed the explicit solutions of the cubic spline interpolation problems. We are now concerned with quintic spline functions. Let 5B[0, n] denote the class of quintic spline functions S(x) defined in the interval [0, n] and having the points 0, 1, • • • , n — 1 as knots. This means that the restriction of S(x) to the interval (*>, J > + 1 ) (P = 0, • • • , n — \) is a fifth degree polynomial, and that S(x)£;C[0, n]. With these functions we can solve uniquely the following three types of interpolation problems. 1. Natural quintic spline interpolation. We are required to find S(x) G*55[0, n] such as to satisfy the conditions