  Starting with the Fourier integral theorem, we present natural Monte Carlo
estimators of multivariate functions including densities, mixing densities,
transition densities, regression functions, and the search for modes of
multivariate density functions (modal regression). Rates of convergence are
established and, in many cases, provide superior rates to current standard
estimators such as those based on kernels, including kernel density estimators
and kernel regression functions. Numerical illustrations are presented.
