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 ∑kdj,kψj,k ∈ L2(ℝ) the function series ft whose wavelet coefficients are dtj,k = dj,k1, 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 ∈ L2(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)