Asymptotic Expansions of Multiple Integrals of Rapidly Oscillating Functions

Expansions of multiple integrals w(aixi,.. .,anxn)g(xi,... ,x„)dxi ■ ■ -dxn, r°i r°n I " Ja\ J a„ where w is a function on R" which is (bk oj,)-periodic in the fcth variable, k = 1,... ,n, and g is smooth, are given in terms of negative powers of the integers o\,.. .,an. Estimates of the remainder term in the expansion are also given. Introduction. Let w be a function with integrable pth power and g a smooth function on an interval Q in R". Denote by w the periodic extension of w to R". Set ox = ( w(ax) is a rapidly oscillating function. Our purpose is to obtain an expansion of the integral r(w;g) / w(ax)g(x) dx JQ in terms of negative powers of o\,... ,on. We recall the standard result, cf. [3], characterizing the asymptotic behavior of Irr{w; g) as lim / w(cTx)g(x) dx = I -rrr. / w(x) dx) I g(x) dx *i-+ooJQ \\Q\ JQ ) \JQ