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The Frisch-Parisi conjecture I: Prescribed multifractal behavior, and a partial solutio

Abstract

In this work and its companion, we construct Baire function spaces in which typical elements share the same prescribed multifractal behavior and obey a multifractal formalism, providing a solution to the so-called Frisch-Parisi conjecture for functions, an inverse problem raised by S. Jaffard. In this first part, a family~$\mathscr{E}_d$ of almost-doubling fully supported capacities on $\R^d$ with prescribed singularity spectra is constructed. With each $\mu\in \mathscr{E}_d$ we associate a Baire function space $\boldsymbol{B}^{\mu}(\R^d)$ (a generalisation of H\"older-Zygmund spaces) in which typical functions share the same singularity spectrum as $\mu$. This yields a partial solution to the conjecture. In~\cite{BS-FP-2}, we introduce and study a family $\boldsymbol{B}=\{\boldsymbol{B}^{\mu,p}_{q}(\R^d)\}_{\mu\in\mathscr E_d, (p,q)\in[1,+\infty]^2}$ of \textit{heterogeneous} Besov spaces that contains $\{\boldsymbol{B}^{\mu}(\R^d)\}_{\mu\in\mathscr E_d}$ and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside~$\boldsymbol{B}$.
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Dates and versions

hal-02899957 , version 1 (15-07-2020)
hal-02899957 , version 2 (23-05-2023)

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Julien Barral, Stephane Seuret. The Frisch-Parisi conjecture I: Prescribed multifractal behavior, and a partial solutio. 2020. ⟨hal-02899957v1⟩

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