arXiv subjects
Publications and source records attributed to Quentin Rible.
In this article, we prove that, under some assumptions on the so-called environment, prevalent functions in inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023 are multifractal, with a singularity spectrum that we determine. This completes the previous Baire generic results already obtained.
This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B $ξ$ p,q , where $ξ$ is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B $ξ$ p,q along affine hyperplanes belong to another inhomogeneous Besov space. Furthermore, we derive an upper bound for the singularity spectrum of the traces of all functions in B $ξ$ $\infty$,q . This bound is then refined for a prevalent set of functions in B $ξ$ $\infty$,q , for which we explicitly compute the singularity spectrum of their traces. Notably, our analysis reveals that the regularity properties of these affine traces are highly sensitive to the choice of the hyperplane along which the trace is taken.
We prove that, given a wavelet $ψ$, it is possible to choose some multi-integers $(p_j=(p_{j,1},...,p_{j,d}))_{j \in \mathbb{Z}} \in \mathbb{Z}^d$ such that, for every $x=(x_1,...,x_d) \in \mathbb{R}^d$, for infinitely many integers $j$, the tensorized wavelet $\prod_{i=1}^d ψ(2^j x_i-p_{j,i})$ does not vanish at $x$. This non-vanishing property is essential for analyzing some generic regularity properties in certain Sobolev and Besov spaces. The proof relies on an assumption regarding the zeros of $ψ$, which we numerically verify for the first Daubechies wavelets.