Tricki

Littlewood-Paley heuristic for derivative

Quick description

The effect of a derivative on a function f is to accentuate the high frequencies and diminish the low frequencies.

Prerequisites

Fourier transform

Example 1

Suppose that we want to study the rate of change of a typical that is not differentiable eg. , or not even weakly differentiable eg. Weierstrass function.

General discussion

The Littlewood–Paley theory states that one can decompose a function (and even Schwartz space) as

where the blocks are localized in Fourier space on the kth dyadic annulus i.e. with . One can show that for we have

So we have the heuristic relationship .