Local Regularity of Nonsmooth Wavelet Expansions and Application to the Polya Function

We give criteria of pointwise regularity for expansions on Haar or Schauder basis (or spline-type wavelets) corresponding to large Hölder exponents. As an application, we determine the exact Hölder regularity of the Polya function at every point and show that it is multifractal.     

Characterization of Pointwise Hölder Regularity

We study different characterizations of the pointwise Hölder spacesC^s(x₀), including rate of approximation by smooth functions and iterated differences. As an application of our results we study the class of functions that are Hölder exponents and prove that the Hölder exponent of a continuous function is the limit inferior of a sequence of continuous functions, thereby improving a theorem of S. Jaffard.     

Self-Affine Fractal Functions and Wavelet Series

We consider functions represented by series ∑_g  G c_gψ(g ^− 1(x)) of wavelet-type, where G is a group generated by affine functions L₁,…,L_n and ψ is piecewise affine. By means of those functions we characterize the class of self-affine fractal functions, previously studied by Barnsley et al. We compute their global and local Hölder exponents and investigate points of non-differentiability. Wavelet-representations for various continuous nowhere differentiable and singular functions are presented. Another application is the construction of functions with prescribed local Hölder exponents at each point.     

The local Hölder function of a continuous function

This work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0.     

Wavelets Techniques for Pointwise Anti-H?lderian Irregularity

In this paper, we introduce a notion of weak pointwise H?lder regularity, starting from the definition of the pointwise anti-H?lder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise H?lder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (referred to here as strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum differs from the strong one.     

A Family of Functions with Two Different Spectra of Singularities

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Hölder exponent of these functions and also their local \(L^p\) regularity, computing the so-called \(p\) -exponent. We prove that in the general case the Hölder and \(p\) -exponent are different at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Hölder and \(L^p\) local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family.     

Global Minimization Algorithms for Holder Functions

This paper deals with the one-dimensional global optimization problem where the objective function satisfies a Hölder condition over a closed interval. A direct extension of the popular Piyavskii method proposed for Lipschitz functions to Hölder optimization requires an a priori estimate of the Hölder constant and solution to an equation of degree N at each iteration. In this paper a new scheme is introduced. Three algorithms are proposed for solving one-dimensional Hölder global optimization problems. All of them work without solving equations of degree N. The case (very often arising in applications) when a Hölder constant is not given a priori is considered. It is shown that local information about the objective function used inside the global procedure can accelerate the search signicantly. Numerical experiments show quite promising performance of the new algorithms.     

Baire generic histograms of wavelet coefficients and large deviation formalism in Besov and Sobolev spaces

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. Firstly, we compute generically (in the sense of Baire's categories) these histograms in Besov and Lp,s(T) spaces, where T is the torus Rd/Zd (resp. in the Baire's vector space where s:q?s(q) is a C¹ and concave function on R⁺ satisfying 0?s^′?d and s(0)>0). Secondly, as an application, we deduce some extra generic properties for the histograms in these spaces, and study the generic validity of the large deviation multifractal formalism in Besov and Lp,s spaces for s>d/p (resp. in the above space V).     

Stability of Calderón inverse conductivity problem in the plane

It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable when the conductivities are Hölder continuous with any exponent α>0.     

On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

We study the Dirichlet problem for the parabolic equation u_t = Δu^m, m > 0, in a bounded, non-cylindrical and non-smooth domain Ω ^{N + 1}, N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent is critical as in the classical theory of the one-dimensional heat equation u_t = u_xx.     

Interpolatory wavelets for manifold-valued data

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.     

Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{?}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis. October 1, 1996. Date revised: May 13, 1997. Date re-revised: January 10, 1998. Date accepted: February 27, 1998.     

Large deviation principle for enhanced Gaussian processes

We study large deviation principles for Gaussian processes lifted to the free nilpotent group of step N. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Hölder- or modulus topology, appears as special case.     

Riesz Bases of Splines and Regularized Splines with Multiple Knots

This paper deals withL₂(R)-norm and Sobolev-norm stability of polynomial splines with multiple knots, and with regularized versions thereof. An essential ingredient is a result on Hölder continuity of the shift operator operating on a B-spline series. The stability estimates can be reformulated in terms of a Riesz basis property for the underlying spline spaces. These can also be employed to derive a result on stable Hermite interpolation on the real line. We point to the connection with the problem of symmetric preconditioning of bi-infinite interpolation matrices.     

A Theorem on Multiplicators in Nonhomogeneous Hölder Spaces and Some of Its Applications

In Sec. 2, sufficient conditions of the boundedness of convolution operators with kernel m in a nonhomogeneous Hölder space are given in terms of the Fourier-image of m. After that, the results of Sec. 2 are used to prove the solvability in a Hölder space of the Cauchy problem for linear systems of the hydrodynamic type. Bibliography: 6 titles.     

Prevalence of Multifractal Functions in S<Superscript>v</Superscript> Spaces

Spaces called S^v were introduced by Jaffard [16] as spaces of functions characterized by the number ≃ 2^ν(α)j of their wavelet coefficients having a size ≳ 2^−αj at scale j . They are Polish vector spaces for a natural distance. In those spaces we show that multifractal functions are prevalent (an infinite-dimensional “almost-every”). Their spectrum of singularities can be computed from ν, which justifies a new multifractal formalism, not limited to concave spectra.     

A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics

We establish a version of the Grobman-Hartman theorem in Banach spaces for nonuniformly hyperbolic dynamics. We also consider the case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. More precisely, we consider sequences of Lipschitz maps Am+fm such that the linear parts Am admit a nonuniform exponential dichotomy, and we establish the existence of a unique sequence of topological conjugacies between the maps Am+fm and Am. Furthermore, we show that the conjugacies are Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. To the best of our knowledge this statement appeared nowhere before in the published literature, even in the particular case of uniform exponential dichotomies, although some experts claim that it is well known in this case. We are also interested in the dependence of the conjugacies on the perturbations fm: we show that it is Hölder continuous, with the same Hölder exponent as the one for the conjugacies. We emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial. In particular, we need to consider several additional Lyapunov norms.     

Detecting and creating oscillations using multifractal methods

By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f, we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ∑_j ∑_k d_j,k ψ _j,k ∈ L ²(?) the function series f^t whose wavelet coefficients are d ^t _j,k = d_j,k 1 , for some fixed real number γ > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series f^t may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f ∈ L ²(?) such that the associated thresholded function series f^t effectively possesses oscillating singularities which were not present in the initial function f . This series f^t is a typical example of function with homogeneous non‐concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)