Abstract
In this paper we develop quadrature formulas for splines with equispaced knots. For special classes of splines, we give simple explicit expressions for the weights in the quadrature formulas in terms of the zeros of the Euler-Frobenius polynomials and show that these weights are positive. The zeros of these polynomials of odd degree up to 15, are given by Nilson [2] and by Schoenberg and Silliman [4] to a high degree of accuracy. The general quadrature formulas can also be used to obtain the cubic natural spline quadrature formula given in Ahlberg, Nilson and Walsh [ 1, pp. 44-471 and the semicardinal odd order natural spline formula of Schoenberg and Silliman [4]. Two of the quadrature formulas that we derive can be stated as follows. Let S be a spline of odd degree m with knots at the integers 0, l,..., n. Suppose that S’*“(O+) = S’*“(C) = 0, for 2 < 2i < m 1, where ,S”’ denotes the kth derivative of S. Then
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