Wavelet expansions, function spaces and multifractal analysis

The purpose of this tutorial is to describe the interplay between three subjects: function spaces, wavelet expansions, and multifractal analysis. Some relationships are now classical. Wavelet bases were immediately considered as remarkable by analysts because they are unconditional bases of ‘most’ function spaces. This property is a key feature of the denoising algorithms of Donoho, for instance. multifractal analysis tries to derive the Hausdorff dimensions of the Holder singularities. Wavelet techniques proved the most efficient tool in the numerical computation of the spectra of singularities of turbulent flows. Our purpose is first to present these points, and then to show how ideas have developed in the recent interplay between these three fields. Refinements of the numerical techniques introduced to compute turbulence spectra have led to the introduction of new function spaces, which turn out to be the right setting to determine the fractal dimensions of graphs, and offer natural extensions of the Besov spaces to negative p’s. The ‘function space setting’ allows one to derive Baire-type results for the value of spectra. Keeping the histograms of wavelet coefficients gives richer information than just keeping the moments of these histograms (which corresponds to keeping only the knowledge of the function spaces to which the function belongs). We compare the probabilistic results (obtained from histograms) with the above Baire-type results.