CLASSICAL INTEGRATION RULES AND WIENER MEASURE

AMARJIT S. BUDHIRAJA ANDREW J. SOMMESE

Abstract. 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 f^{(j)}(0) = 0 for j ≤ N. Let μ_{N} denote 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} < ⋯ < x_{K} = 1 and weights w := (w_{ij}; i = 0, ⋯ K; j = 0, ⋯, N) define the integration

rule f → A_{w}(f) := ∑_{i,j} w_{i,j}f^{(j)}(x_{i}), to approximate I(f) := ∫_{0}^{1} f(x) dx. A study

is made of the average square error ∫_{C_{0}^{N}} |I(f) − A_{w}(f)|^{2} dμ_{N}(f); for what weights

w the minimum is achieved; for what sets of nodes this minimum is minimized; and what the minimum is in this case.