Algorithm for the computation of the coefficients of powers of polynomials

Abstract

One of the approaches to determine the global maximum of a multivariate function f(x) within a "feasible region" R in the Euclidean n-space is based on the evaluation of the so-called functional moments of f(x), that is, the integrals I[subscript k] = [integral][subscript R]f(x)[superscript k]dx for a sequence of integral k. This dissertation is concerned with algorithms accomplishing this task in three special cases. The first case arises when f(x) is a multivariate polynomial and R is the n dimensional hypercube. In the second case, f(x) is a multivariate expansion into trigonometric functions and region R is the hypercube. Finally, a third case is considered where f(x) is given by a multivariate polar expansion and R is a smooth convex region in the sense that the distance from an origin in the interior of R to the boundary is a low degree trigonometric expansion in the space angles. The application of I[subscript k] to nonconvex programming is not spelled out in this dissertation but a number of generalizations of the above problem, useful in mathematical programming, are also treated.