Oscillatory Integrals

人◕ ‿‿ ◕人 *Whale face*

Whale hello there! I recently had a whale of a time (am I using that phrase correctly?) reading about oscillatory integrals and I wanted to share some notes and brief thoughts.

First the notes. My two of my favorite references that I found are:

  1. Terry Tao’s Fourier Analysis Notes, specifically notes 8 on the Stationary Phase. Please see here
  2. Stein’s Book on Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. But specifically chapter 8: Oscillatory Integrals of the first kind. (Sorry I don’t have a reference for this one)

My key takeaway was that oscillatory integrals are integrals with an oscillation term such as e^{i \lambda x} or even more generally a(x)e^{i \lambda \phi (x)}

Where a(x) is an amplitude function and \phi (x) is called the phase function – and it controls something about the direction and rate of the oscillation.

To get an estimate of simple versions of these integrals people resort to one main tool: “The method/ principle of the non-stationary phase”

In one dimension, the idea is that if your phase function (and amplitude if you are using one) is smooth with non-zero, lower bounded derivatives (i.e. non-stationary) then the integral is asymptotically bounded by something like ~O(\lambda^{-N}) for all N. (Think of the Rieman-Lesbesgue Lemma)

And if there is a degeneracy, say if all the conditions hold but at the kth derivative the phase function is 0, then the integral is asymptotically ~O(\lambda^{-1/k}) and all the higher order terms of this can even be explicitly written down by formula.

In effect, by tricks which look suspiciously like integration by parts, we can “trade” regularity on our phase function for better decay on our integrals!

Side Remarks:

  • The theory changes a bit for higher dimensions
  • In Stein’s book, under further results A, Stein discusses how the amplitude affects his calculations. Terry does this discussion too in his notes, but with Stein you have to check the “Further Results” section.






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