The notion of gentle spaces, introduced by Jaffard, describes what would be an “ideal” function space to work with wavelet coefficients. It is based mainly on the separability, the existence of bases, the homogeneity, and the γ-stability. We prove that real and complex interpolation spaces between two gentle spaces are also gentle. This shows the relevance and the stability of this notion. We deduce that Lorentz spaces  and  spaces are gentle. Further, an application to nonlinear approximation is presented.

he study of d dimensional traces of functions of m several variables leads to directional behaviors. The purpose of this paper is two-fold. Firstly, we extend the notion of one direction pointwise Hölder regularity introduced by Jaffard to multi-directions. Secondly, we characterize multi-directional pointwise regularity by Triebel anisotropic wavelet coefficients (resp. leaders), and also by Calderón anisotropic continuous wavelet transform.

Many natural mathematical objects, as well as many multi-dimensional signals and images from real physical problems, need to distinguish local directional behaviors (for tracking contours in image processing for example). Using some results of Jaffard and Triebel, we obtain criteria of directional and anisotropic regularities by decay conditions on Triebel anisotropic wavelet coefficients (resp. wavelet leaders).

We study the de Rham function: the unique continuous (nowhere differentiable) function  with  satisfying the functional equation .

We characterize pointwise directional regularity by highly oriented multi-scaled wavelet coefficients.

In this paper, we prove that anisotropic homogeneous Besov spaces  are gentle spaces, for all parameters s,p,q and all anisotropies . Using the Littlewood–Paley decomposition, we study their completeness, separability, duality and homogeneity.

Histograms of wavelet coefficients are expressed in terms of the wavelet profile and the wavelet density. The large deviation multifractal formalism states that if a function f has a minimal uniform Hölder regularity then its Hölder spectrum is equal to the wavelet density. The purpose of this paper is twofold.

The oscillation spaces Os,s′p(Rd)Ops,s′(Rd) introduced by Jaffard are a variation on the definition of Besov spaces for either s ≥ 0 or s ≤ −d/p. On the contrary, the spaces Os,s′p(Rd)Ops,s′(Rd) for −d/p < s < 0 cannot be sharply imbedded between Besov spaces with almost the same exponents, and, thus, they are new spaces of really different nature. Their norms take into account correlations between the positions of large wavelet coefficients through the scales.