Related papers: A generalization of the Takagi function for beta-e…

We investigate the H\"older regularity of the function $T$ of the probability of tending to one minimal set, the partial derivatives of $T$ with respect to the probability parameters, which can be regarded as complex analogues of the Takagi…

In this paper, we study the uniform H\"older continuity of the generalized Riemann function $R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad…

Let $\beta>1$ be a non-integer. First we show that Lebesgue almost every number has a $\beta$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $\beta$-expansions of the same given frequency.…

We consider the random iteration of finitely many expanding $\mathcal{C}^{1+\epsilon}$ diffeomorphisms on the real line without a common fixed point. We derive the spectral gap property of the associated transition operator acting on…

This paper gives the pointwise H\"older (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These…

A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is…

The classical Hahn-Banach theorem is based on a successive point-by-point procedure of extending bounded linear functionals. In the setting of a general metric domain, the conditions are less restrictive and the extension is only required…

This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $\beta$-expansions. More precisely, let $([0,1),T_{\beta})$ be the $\beta$-dynamical system for a general $\beta>1$ and…

Let (X,d) be a metric space and $ \alpha > 0 $. In this paper, we study extensions of some complex-valued Lipschitz functions, from some special subset $ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the smallest…

In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\beta\in(1,1.787\ldots)$ then every…

We study the sets DF({\beta}) of digit frequencies of {\beta}-expansions of numbers in [0,1]. We show that DF({\beta}) is a compact convex set with countably many extreme points which varies continuously with {\beta}; that there is a full…

Simple asymptotic expansions for the Jacobi functions $P_\nu^{(\alpha, \beta)}(z)$ and $Q_\nu^{(\alpha, \beta)}(z)$ for large degree $\nu$, with fixed parameters $\alpha$ and $\beta$, are surprisingly rare in the literature, with only a few…

In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the…

In this paper we study the set of digit frequencies that are realised by elements of the set of $\beta$-expansions. The main result of this paper demonstrates that as $\beta$ approaches $1,$ the set of digit frequencies that occur amongst…

In this paper, we present an extension of Mittag-Leffler function by using the extension of beta functions (\"{O}zergin et al. in J. Comput. Appl. Math. 235 (2011), 4601-4610) and obtain some integral representation of this newly defined…

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of…

We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices…

We study non-standard number systems with negative base $-\beta$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-\beta}$ of the interval $\big[-\frac{\beta}{\beta+1},\frac{1}{\beta+1}\big)$ into itself, we suggest…

Given $\beta>1$ and $\alpha\in[0,1)$, let $T_{\beta, \alpha}(x)=\beta x+\alpha\pmod 1$. Then under the map $T_{\beta,\alpha}$ each $x\in[0,1]$ has an \emph{intermediate $\beta$-expansion} of the form…

For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be the assertion that for every Dirichlet character $\chi$ and all zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular, ${\rm GRH}[\frac12]$ is…