Pointwise regularity associated with function spaces and multifractal analysis

Abstract

Abstract. The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f ; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces C α E (x0) are constructed, leading to a notion of pointwise regularity with respect to E; the case E = L ∞ corresponds to the usual Hölder regularity, and E = L p corresponds to the T p α (x0) regularity of Calderón and Zygmund. We focus on the study of the spaces T p α (x0); in particular, we give their characterization in terms of a wavelet basis and show their invariance under standard pseudodifferential operators of order 0.

Function Spaces Vs. Scaling Functions: Tools for Image Classification

“…Therefore it plays a similar role as the computation of H min f when dealing with the multifractal analysis based on wavelet leaders. Note that, as above, a spectrum can be attached to the q-exponent, and a multifractal formalism worked out, using the usual procedure; this spectrum is obtained as a Legendre transform of the q-scaling function, see [28,29,32]. An important open question is to understand, for a given function f , the relationships that exist between these q-spectra, as a function of the parameter q.…”

p-exponent and p-leaders, Part I: Negative pointwise regularity

“…In this context, as a possible alternative to (fractional) integration, we propose the use of p-exponents, which potentially take negative values and hence permit to characterize negative local regularity. Though introduced in the theoretical context of PDEs as early as 1961 for p > 1 by Calderón and Zygmund [28], p-exponents were not used in signal processing until the 2000s when their wavelet characterization was proposed [29,30,31]. In the present contribution, we study which information on the local behavior of a function near a singularity is supplied by the knowledge of the collection of p-exponents.…”

Multifractal Analysis Based on p-Exponents and Lacunarity Exponents

“…Note that the support of F α ω,γ is U ω,γ so that the accessibility exponent at 0 of this support is given by (14). The function F α ω,γ is locally bounded if and only if α ≥ 0.…”

“…The case p < 1. The standard way to perform this extension is to consider exponents in the setting of the real Hardy spaces H p (with p < 1) instead of L p spaces, see [14,15]. First, we need to extend the definitions that we gave to the range p ∈ (0, 1].…”