Detecting and creating oscillations using multifractal methods

Abstract

AbstractBy 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 dj,k ψ j,k ∈ L 2(ℝ) the function series ft whose wavelet coefficients are d t j,k = dj,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 ft may have a multifractal spectrum with a support larger than the one of f .We exhibit an example of function f ∈ L 2(ℝ) such that the associated thresholded function series ft effectively possesses oscillating singularities which were not present in the initial function f . This series ft 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)