Section: New Results
Limiting behaviour of wavelets and singularity exponents
The accurate computation of singularity exponents is a key aspect of the Microcanonical Multifractal Formalism (MMF). The singularity exponents are computed using a wavelet projection of a multifractal measure: the wavelet projection of measure at x = 0 and at scale r in is given by:
with wavelet and
Therefore, it is important to understand, both qualitatively and quantitatively, the influence of the wavelet on the computation of the singularity exponent at x = 0 .
We study the influence of the wavelet's tail at infinity on the singularity exponent computed at x = 0 . Using the Lebesgue density theorem and the analysis of the integral appearing in (2 ), we obtain that the local behaviour of the wavelet at x = 0 lets the singularity exponent at 0 unchanged when r0 , as long as (0)0 , but if the wavelet decreases towards infinity according to a power law : when , powers of n0 pop up in the integral which may change the behaviour of 's singularity exponent at 0 , when r tend towards 0. This qualitative analysis is the first step towards a mathematical understanding of numerical artefacts appearing in the algorithms implementing the wavelet projection. The second step will consist in computing the quantitative dependance of the wavelet's tail on the singularity exponents.