Asymptotic expansions of multiple integrals of rapidly oscillating functions

Abstract:

Expansions of multiple integrals \[ \int _{{a_1}}^{{b_1}} \cdots \int _{{a_n}}^{{b_n}} {w({\sigma _1}{x_1}, \ldots ,{\sigma _n}{x_n})g({x_1}, \ldots ,{x_n})\;d{x_1} \cdots d{x_n},} \] where w is a function on ${{\mathbf {R}}^n}$ which is $({b_k} - {a_k})$-periodic in the kth variable, $k = 1, \ldots ,n$, and g is smooth, are given in terms of negative powers of the integers ${\sigma _1}, \ldots ,{\sigma _n}$. Estimates of the remainder term in the expansion are also given.