Tricki
a repository of mathematical know-how

Square and rearrange

Note iconIncomplete This article is incomplete. This article needs more work

Quick description

To control an integral \int_E f(x)\ d\mu(x) or a sum \sum_{n \in A} f(n), take its magnitude squared, expand it into a double integral \int_E \int_E f(x) \overline{f(y)}\ d\mu(x) d\mu(y) or a double sum \sum_{n \in A} \sum_{m \in A} f(n) \overline{f(m)}, and then rearrange, for instance by making the change of variables y=x+h or n=m+h.

This often has the effect of replacing a phase e^{2\pi i \phi(n)} or e^{2\pi i \phi(x)} in the original integrand by a "differentiated" phase such as e^{2\pi i (\phi(n+h)-\phi(n))} or e^{2\pi i (\phi(x+h)-\phi(x))}. Such differentiated phases are often more tractable to work with, especially if \phi had a "polynomial" nature to it.

Prerequisites

harmonic analysis, analytic number theory

Example 1

This is a classic example: to compute the integral = \int_{-\infty}^\infty e^{-\pi x^2}\ dx, square it to obtain

 A^2 = \int_{\R^2} e^{-\pi(x^2+y^2)}\ dx dy

then rearrange using polar coordinates to obtain

 A^2 = \int_0^2\pi \int_0^\infty e^{-\pi r^2} r\ dr d\theta.

The right-hand side can easily be evaluated to be 1, so the positive quantity = \int_{-\infty}^\infty e^{-\pi x^2}\ dx must also be 1.

Example 2

(Gauss sums)

Example 3

(Weyl sums)

Example 4

(The TT^* method, say to obtain Hormander's L^2 oscillatory integral estimate)

Example 5

(The large sieve)

General discussion

A variant of this trick is the van der Corput lemma for equidistribution.

Comments

Post new comment

(Note: commenting is not possible on this snapshot.)

Before posting from this form, please consider whether it would be more appropriate to make an inline comment using the Turn commenting on link near the bottom of the window. (Simply click the link, move the cursor over the article, and click on the piece of text on which you want to comment.)

snapshot
Notifications