A Bridge Between Geometric Measure Theory and Signal Processing: Multifractal Analysis

p-exponent and p-leaders, Part I: Negative pointwise regularity

“…When α < 1, the Taylor polynomial boils down to a constant P x0 (x) ≡ X(x 0 ). A more general discussion on P x0 will be given in Section 2.2 (see also [32]). …”

“…Appendix A). This implies that by picking up the integer part of h p (x 0 ) we get the polynomial P which corresponds to the largest possible value of α. Theorem 1 (Point 2) below indicates that one can fix a unique Taylor polynomial of X at x 0 , whose coefficients are independent of p, and are referred to as the (generalized) Peano derivatives of X at x 0 [33,32]. One of the main advantages of the wavelet framework developed in Section 3 is however that the computation of P x0 is not required to measure the p-exponent.…”

“…One of the main advantages of the wavelet framework developed in Section 3 is however that the computation of P x0 is not required to measure the p-exponent. The Taylor polynomial is thus not further discussed and readers are referred to e.g., [32].…”

“…Note the use of an L 1 normalization for the wavelet coefficients that better fits local regularity analysis and yields the correct self-similarity exponent of the wavelet coefficients for self-similar functions, see [18,16,24,32]. For further details on wavelet bases and wavelet transforms, the reader is referred to e.g., [37].…”

“…Wavelet coefficients can furthermore be used to assess whether X locally belongs to L p or not, i.e., to verify the condition that is a priori required for using the corresponding p-exponent [26,32,27]. Let S c (j, p) denote the wavelet structure function, defined as the space/time averages of the magnitude of wavelet coefficients raised to a positive power p > 0:…”