Function series with multifractal variations

In this paper, we study three classes of multifractal function series using a work achieved for a new class of measures defined in [3]. The originality of these function series consists in the fact that the sizes of the jumps (or of the amplitudes of the pulses) depend on the location of the jump points and on a measure mu. In particular, there may be a strong heterogeneity in the distribution of the size of the jumps. These function series f are defined by f(X) = Sigma(jgreater than or equal to1) 1/j(2) Sigma(b = 0)(bj-1) mu([kb(-j),(k + 1)b(-j)))V-j,V- k(X), where psi(j,k) is a contracted and dilated version of a single function psi. This function psi will either be a wavelet, a pulse, or a piecewise linear function. We show that under suitable conditions on the measure mu, the multifractal spectrum of f can be computed. For large classes of measures, the spectrum is linear between 0 and a critical value h(C), and if h greater than or equal to h(C), f and mu share the same spectrum. This untypical shape is the result of the combination of the multifractal measure mu with the rapid variations or discontinuities of the functions psi(j, k). (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.