The Dimension Distribution and Quadrature Test Functions

This paper introduces the dimension distribution for a square integrable function f on [0, 1]. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically. Many commonly considered quadrature test functions are sums or products of univariate functions, and as a result have particularly simple dimension distributions. Recently some high dimensional isotropic integrals have been successfully treated by quasi-Monte Carlo methods. We show numerically that one such function in 25 dimensions is very nearly a superposition of functions of 3 or fewer variables, explaining the success of QMC on that problem. A new result shows that certain isotropic polynomials of degree n generate integrands that are exact superpositions of functions of n or fewer variables.