Classical integration rules and Wiener measure
Amarjit S. Budhiraja and Andrew J. Sommese
In this paper precise calculations are made of the square of the error of classical integration rules averaged over spaces of differentiable functions. Precisely, let $C_0^N$ denote the space of N-times continuously differentiable functions on [0,1] with all derivatives up to and including the N-th, 0 at 0. On this space put the measure induced by the N-times integrated Wiener process on $C_0^N$. For a given set of nodes $0=x_0< x_1<\cdots <x_K=1$ and weights ${\bf w} := (w_{ij}; \;\; i=0, \cdots K; j = 0, \cdots , N)$ define the integration rule $f\to A_{{\bf w}}(f) := \sum_{i,j} w_{i,j}f^{(j)}(x_i),$ to approximate $\int_0^1f(x)\,dx$. A study is made of the average square error over $C_0^N$; for what weights ${\bf w}$ the minimum is achieved; for what sets of nodes this minimum is minimized; and what the minimum is in this case.