The Integration and DREs of Rapidly Oscillating Functions

In this chapter, we will discuss the following n-dimensional oscillatory integral $$ \int_0^1 {f\left( {{x_1}, \ldots ,{x_n};\left\langle {{N_1}{x_1}} \right\rangle , \ldots ,\left\langle {{N_n}{x_n}} \right\rangle } \right)d{x_1} \ldots d{x_n}} $$ where, for all i = 1…, n, ‹N i x i ›= N i x i — [N i x i ], the fractional part of N i x i , and N i ≥ 2 are large positive integers. Here, function f in the integral is called a rapidly oscillating function. In this chapter, we will show that an integral of a continuous function f (x1,…, x n ; y1,…, y n ) over a 2n-dimensional sphere or cube can be approximated by a sequence of oscillatory integrals. In particular, if the continuous function f is also periodic in terms of each independent variable in a subset consisting of any n — 2 of 2n variables, then the corresponding 2n-dimensional integral can be reduced to a one-dimensional oscillatory integral with, of course, a remainder. Hence, we need to give approximation expansions of one-dimensional oscillatory integrals. A basic expansion formula is shown as follows (see Section 4).