Related papers: Level sets of prevalent Weierstrass functions

We study the level sets of prevalent H\"older functions. For a prevalent $\alpha$-H\"older function on the unit interval, we show that the upper Minkowski dimension of every level set is bounded from above by $1-\alpha$ and Lebesgue…

In this paper we consider functions of the type $$f(x) = \sum_{n=0}^\infty a_n g(b_nx+\theta_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \geq b >1$, $a^2b> 1$…

For a real analytic periodic function $\phi:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $\lambda\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function…

Let $W_{\lambda,b}(x)=\sum_{n=0}^\infty\lambda^n g(b^n x)$ where $b\geqslant2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Bar\'ansky, B\'ar\'any and Romanowska (2013) and…

This paper examines dimension of the graph of the famous Weierstrass non-differentiable function \[ W_{\lambda, b} (x) = \sum_{n=0}^{\infty}\lambda^n\cos(2\pi b^n x) \] for an integer $b \ge 2$ and $1/b < \lambda < 1$. We prove that for…

We show that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log\lambda/\log b$, for every integer $b\ge 2$ and every $\lambda\in (1/b,1)$. Replacing $\cos(2\pi x)$…

We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + \theta_n)$, where $g$ is a periodic Lipschitz real function and…

Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by \[ f_{\Theta,\Lambda}(t)= \left(…

In this paper we study the set of functions $\GG$-valued which can be approximated by $\GG$-valued continuous functions in the norm $L^\infty_{\GG}(I,w)$, where $I$ is a compact interval, $\GG$ is a real and separable Hilbert space and $w$…

For a real analytic periodic function $\phi:\mathbb{R}\to \mathbb{R}$, an integer $b\ge 2$ and $\lambda\in (1/b,1)$, we prove the following dichotomy for the Weierstrass-type function $W(x)=\sum\limits_{n\ge 0}{{\lambda}^n\phi(b^nx)}$:…

We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive samples-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of…

Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq…

A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first…

We study H\"older continuity, $p^\mathrm{th}$-variation function and Riesz variation of Weierstrass-type functions along the sequence of $b$-adic partitions, where $b>1$ is an integer. By a Weierstrass-type function, we mean that in the…

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the…

We study several fractal properties of the Weierstrass-type function \[ W(x)=\sum_{n=0} ^\infty \lambda (x) \lambda(\tau x) \cdots \lambda (\tau ^{n-1}x)\, g(\tau ^n x), \] where $\tau :[0,1)\to[0,1)$ is a cookie cutter map with possibly…

Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In…

We consider the escaping parameters in the family $\beta\wp_\Lambda$, i.e. these parameters for which the orbits of critical values of $\beta\wp_\Lambda$ approach infinity, where $\wp_\Lambda$ is the Weierstrass function. Unlike to the…

We consider the Banach space consisting of real-valued continuous functions on an arbitrary compact metric space. It is known that for a prevalent (in the sense of Hunt, Sauer and Yorke) set of functions the Hausdorff dimension of the image…

A consistent notation for the Weierstrass elliptic function $\wp(z;g_{2},g_{3})$, for $g_{2} > 0$ and arbitrary values of $g_{3}$ and $\Delta \equiv g_{2}^{3} - 27 g_{3}^{2}$, is introduced based on the parametric solution for the motion of…