Generalized spaces of pointwise regularity: toward a general framework for the WLM

Abstract

In this work we generalize the spaces T u p introduced by Calderón and Zygmund using pointwise conditions emanating from generalized Besov spaces. We give conditions binding the functions belonging to these spaces and their wavelet coefficients. Next, we propose a multifractal formalism based on such spaces which generalizes the so-called wavelet leaders method and show that it is satisfied on a prevalent set.

Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions

“…Wavelet basis and biorthogonal wavelet basis give a powerful tool to study the regularity and irregularity of functions or signals belonging to numerous functional spaces, see e.g. [21,31,6,19,1,20,12,17,25,27]. In this paper, they will be used to get irregularity properties in Section 3.…”

“…while, for all ε > 0, f h / ∈ C h+ε (t) and so, again, h f h (t) = h. In fact, the finiteness of the limits ( 8), ( 10) and (12) means that the function f belongs to a generalized pointwise Hölder space [25,27] associated to the corresponding modulus of continuity.…”

“…for some positive constant C by assumption (27). In order to deal with the second term, we use in addition the fast decay of the wavelet to get the existence of a natural number N and a constant C such that…”

“…The aim of this paper is to characterize more precisely the pointwise behavior of such a wavelet series, in the spirit of what is known for the Brownian motion. Recently, generalized Hölder spaces has been introduced to address this kind of questions [25,27] as well as the regularity of solutions of partial differential equations [26]. This article is the continuation of the work done in [2,4,22] for the Brownian motion.…”

Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

“…Remark 2.2. Let us mention that wavelet leaders can not be used to prove the finiteness of the limits in Theorem 1.2 because they do not precisely characterize the pointwise regularity, see for instance [27,31] for more details.…”