Wavelets

Energy Distribution of EEG Signal Components by Wavelet Transform

“…i ED is the energy of the detail at decomposition level i and l EA is the energy of the approximate at decomposition level l Jaffard et al, 2001;Mertins, 1999;Omerhodzic et al, 2008Omerhodzic et al, , 2011. Figure 1 shows the three signals from the analyzed database.…”

Pointwise regularity associated with function spaces and multifractal analysis

“…Note also that, if f satisfies (15), then f ∈ T p α (x 0 ) for any p < ∞. This is in sharp contrast with the MULTIFRACTAL ANALYSIS 107 two-microlocal condition obtained in [8] as a consequence of C α (x 0 ) regularity which does not imply any T p u (x 0 ) regularity result (or even that f locally coincides with a function).…”

“…Conversely, if (14) holds (or if (15) holds in the case p = +∞) and if α / ∈ IN, then f ∈ T p α (x 0 ). This theorem will be proved in two steps.…”

“…We can forget the "low frequency component" of f corresponding to j < 0 in its wavelet decomposition, since its contribution belongs locally to C r (IR d ) (for r-smooth wavelets). Let Λ j denote the set of dyadic cubes of width 2 −j , ∆ j f = λ∈Λj c λ ψ λ , and let P j (x − x 0 ) denote the Taylor polynomial of ∆ j f of degree [α] at x 0 ; (14) or (15) …”

“…It follows that the bound obtained there is Cρ α log(1/ρ). Thus, if α ∈ IN and if (14) holds (or if (15) holds in the case p = +∞) then f satisfies ∃R, C > 0 and a polynomial P of degree less than α such that…”