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 .
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