Differentiability and Dimension of Some Fractal Fourier Series

Vortex filament equation for a regular polygon

“…More concretely, in [32], it was proved that Riemann's non-differentiable function is a multifractal (see also [33], where other relevant trigonometric sums are considered). In other words, it was proved that the set of times t that have the same Hölder exponent is a fractal with a dimension depending on the Hölder exponent, in such a way that the conjecture stated by Frisch and Parisi in [34] is fulfilled (see also [35], for more details at this respect and the connection of this question with fully developed turbulence and intermittency).…”

Automorphic forms and differentiability properties

“…As θ is not cuspidal at i∞ or 0 and cuspidal at 1/2, Corollary 2.2.4 (with α = 1, r = 1/2) implies that F (x) is differentiable in the orbit of 1/2 = {a/(2b) : 2 | a, b} [6] and non-differentiable elsewhere [7]. By Corollary 2.2.3, the value of the derivative (when it exists) is −π [6], and by Theorem 2.3 the fractal dimension of its graph [3] is 5/4. §3.…”

“…)) proves that the series formally defining F (x 0 ) is not even Abel summable for α < 3/2; hence, by III.7.6 in [17], F is not differentiable at x 0 (cf. [3]). This is also the case for k > 1 when α < k/2 + 1, because the expansion at the cusp x 0 implies that the gamma integral representation …”

Multifractal behavior of polynomial Fourier series

“…For instance, it is known that the (box counting) dimension of the graph of R is 5/4, in particular it is a fractal [Cha04], [CC99].…”

“…But it is important to keep in mind that the spectrum of singularities requires to deal with subsets having fractional Hausdorff dimension. Then this average has to be done in restricted sets and, for instance, integration that is useful to compute the dimension of the graph of F [CC99] [CU07] is too coarse here. Once this fine average is carried out, we can construct some fractal sets whose elements have special diophantine approximation properties that allow to extract a main term for the variation of F in some ranges.…”