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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 f(x)\in L^{2} that is not differentiable eg. =|x|1_{[-1,1]}(x), or not even weakly differentiable eg. Weierstrass function.

General discussion

The Littlewood–Paley theory states that one can decompose a function f\in L^{2} (and even Schwartz space) as

f(x)=\sum_{k}P_{k}f(x),

where the blocks P_{k}f(x) are localized in Fourier space on the kth dyadic annulus i.e. \widehat{P_{k}f}(\xi)=\hat{f}(\xi)\phi(\frac{\xi}{2^{k}}) with supp\phi(\xi)\subset \{\frac{1}{2}\leq |\xi|\leq 2 \}. One can show that for p\in [1,\infty] we have

\left \| \nabla P_{k}f \right \|_{L^{p}}\sim 2^{k}\left \|  P_{k}f \right \|_{L^{p}}.

So we have the heuristic relationship \nabla \sim \sum_{k}2^{k}P_{k}.