  Many examples of signals and images cannot be modeled by locally bounded
functions, so that the standard multifractal analysis, based on the H\"older
exponent, is not feasible. We present a multifractal analysis based on another
quantity, the p-exponent, which can take arbitrarily large negative values. We
investigate some mathematical properties of this exponent, and show how it
allows us to model the idea of "lacunarity" of a singularity at a point. We
finally adapt the wavelet based multifractal analysis in this setting, and we
give applications to a simple mathematical model of multifractal processes:
Lacunary wavelet series.
