Abstract
The oscillating multifractal formalism is a formula conjectured by Jaffard expected to yield the spectrum d(h, beta) of oscillating singularity exponents from a scaling function zeta(p, s'), for p > 0 and s' is an element of R, based on wavelet leaders of fractional primitives f-s' of f. In this paper, using some results from Jaffard et al., we first show that zeta(p, s') can be extended on p. R to a function that is concave with respect to p is an element of R and independent on orthonormal wavelet bases in the Schwartz class. We also establish its concavity with respect to s ' when p > 0. Then, we prove that, under some assumptions, the extended scaling function zeta(p, s') is the Legendre transform of the wavelet leaders density of f-s'. Finally, as an application, we study the validity of the extended oscillating multifractal formalism for random wavelet series (under the assumption of independence and laws depending only on the scale).