# Numerical Quadrature of Oscillating Integral With Non-oscillating part

As you will know there are different numerical integrals (I believe Levin's method is the most popular one) for the numerical quadrature of oscillating integrands which may roughly speaking be written as a product of oscillating part and non-oscillating amplitude $exp(i \omega t) A(t)$ (the form can of course be slightly more general). Integrands of the form $exp(i \omega t) A(t)+B(t)$ may be treated using a method for oscillatory and one for non-oscillatory for the first and second part, respectively. I wonder if there are routines also for the efficient quadrature of integrands which cannot be analytically decomposed as $exp(i \omega t) A(t)+B(t)$ but which nevertheless feature a rapidly oscillatory behavior (you can assume fixed frequency) . I.e. they behave like $exp(i \omega t) A(t)+B(t)$ with some unknown $A$ and $B$. It appears to me that I could in principle do better than using a conventional quadrature for non-oscillating integrands as I could try to find $A$ and $B$ using smooth (and successively refining) fits and then run a quadrature for oscillating integrals in order to obtain a numerical estimate for the integral. I wonder if there are any known efficient methods for such problems.

• I don't have an answer, but I suggest you have a look at the publications of Daan Huybrechts on oscillatory integrals. I think he or his group at KULeuven are still quite active in the field. – GertVdE Feb 27 '14 at 9:27
• Your reference has already been helpful. Browsing it I found this math.ethz.ch/~hiptmair/Seminars/OSCINT/OLV06b.pdf which deals with multivariate quadrature of oscillatory integrands. It reminds me to mention that my original integral is similar, a two dimensional one on a certain domain. To perform the first integral which is strictly oscillatory (i.e.) I can use Levin quadrature. The remaining integral is of the form I describe above. So references which deal with 2-dim integrals are also useful. However my integrand can feature integrable singularities. – highsciguy Feb 27 '14 at 18:16
• Some basic literature would be good which explains the terms used in the paper, like 'non-resonance condition' etc. – highsciguy Feb 27 '14 at 18:18