Multifractal analysis on the Heisenberg group
Joint with Stéphane Seuret.
1.1. Sources of multifractal analysis
Multifractal analysis is a toolbox for data analysis (textures, financial or experimental data, diophantine approimation,… see the program of our seminar at UPEC). It introduces classification parameters based on absence of regularity. Why the Heisenberg group ? People from image analysis ask about non isotropic textures.
1.2. Historic examples
Weierstrass’ example of a function which is -Hölder but nowhere differentiable (for every .
Riemann function which is differentiable at infinitely many rational points (but not all of them) and nowhere else,
Its multifractal spectrum was not computed before 1996 (S. Jaffard).
Both examples are self similar. A picture of Weierstrass’ function suggests that irregularity is spread all over. A picture of Riemann’s function looks very different : spikes, points with different left and right derivatives, differentiability point, all over.
Definition 1 The pointwise regularity exponent of at is the largest such that is, up to a polynomial, .
The multifractal spectrum is the function
Example 1 Weierstrass function is monofractal : every point has pointwise regularity exponent .
The multifractal spectrum of Riemann’s function is made of a segment joining and plus a point at .
2. Heisenberg group
The strong anisotropy turns out not the make a big change for the specific questions we solve the existing techniques adapt rather easily. This is why we can state theorems
One can construct smooth, exponentially decaying functions concentrated at , where (which is a subgroup), with vanishing moments, which form a basis of .
Theorem 2 (Global Hölder regularity) Let . A function belongs to iff its -th horizontal derivatives are . Also iff its wavelet coefficients
Theorem 3 (Pointwise Hölder regularity) Let . If a function is at , then
Conversely, if this holds, then is at for all .
2.3. Generic spectrum in H\” older and Besov classes
Theorem 4 Monofractal functions (at ) form a dense subset of .
Definition 5 if
Note that for and . Here, .
Theorem 6 For a dense subset of , the spectrum is a segment between and . For all functions of , the spectrum is below this segment.
Indeed, for the standard example of a Besov function (expressed in wavelet expansion), the pointwise Hölder exponent at is related to the dyadic approximation rate of . Dimensions of isoapproximable sets can be computed. It is a special case of a very general result by Beresnevich, Dickinson and Velani (2006).