The Hausdorff dimension of pulse-sum graphs

Abstract

We consider random functions formed as sums of pulses where Xn are independent random vectors, 0<α<1, and G is an elementary “pulse” or “bump”. Typically such functions have fractal graphs and we find the Hausdorff dimension of these graphs using a novel variant on the potential theoretic method.

A note on the Hausdorff dimension of general sums of pulses graphs

“…The results of the following lemmas are similar to Lemma 3.1, Corollary 3.2 and Corollary 3.3 established in [2].…”

“…The proof is based on the potential theoretic method to calculate the Hausdorff dimension of graphs of many functions, such as the fractional Brownian motion [7] or the random Weierstrass function [6] and those given in the particular case a n = 1 n H and λ n = 1 n , H ∈ (0, 1) in [2]. The potential theoretic ideas are developed in the following section.…”

“…In the same particular case and in higher dimension, the analysis of the fractal sums of pulses has been treated in [3] and [2]. The existence and regularity of functions defined by (1) have been studied in [1].…”

Analysis of the fractal sum of pulses

“…A natural conjecture is that the Hausdorff dimension of Γ F is D + 1 − H almost surely. This has been proved in [00] during the process of revision of this paper.…”

Existence et régularité höldérienne des fonctions de bosses

“…Dans [5] et [4] et dans le cas particulier λ n = 1/n et a n = 1/n H , les auteurs ont montré que la dimension de boîte de la restriction du graphe de F sur [0, 1] vaut presque sûrement 2 − H et que sous des hypothèses supplémentaires sur G la dimension de Hausdorff du graphe de F vaut aussi presque sûrement 2 − H. Nous renvoyons aussià [3] pour un large exposé sur les fonctions de bosses.…”