Most methods for oscillatory integrals I know about deal with integrals of the form $$ \int f(x)e^{i\omega x}\,dx $$ where $\omega$ is large.
If I have an integral of the form $$ \int f(x)g_1(x)\cdots g_n(x)\,dx, $$ where $g_k$ are oscillatory functions whose roots are only known approximately, but some kind of asymptotic form $$ g_k(x) \sim e^{i\omega_k x} $$ is known, with the frequencies $\omega_k$ all different (and $\mathbb{Q}$-linearly independent), then how can I evaluate this integral?
Unlike in the case of $e^{i\omega x}$, the polynomial integrals $\int x^a \prod g_k(x)$ are not known, so I can't construct a set of polynomial interpolants for $f(x)$ and integrate the interpolants exactly.
In my exact problem, $g_k$'s are Bessel functions $J_0(\omega_k x)$, and $f(x)=x^\alpha$, and the region of integration is $[0,\infty)$. The method I am using now is to sum up integral contributions over intervals $[x_{k-1},x_k]$ between roots up to some cutoff $M$, then use the asymptotic expansion for $g_k(x)$ for large $x$. This algorithm's time complexity is exponential in $n$ because it involves expanding the product $g_1\ldots g_n$, each of which has a number $r$ of asymptotic terms, giving $r^n$ total terms; pruning terms that are too small doesn't reduce the run time enough to make this feasible for large $n$.
Heuristic non-rigorous answers, suggestions and references are all welcome.
