Fractal interpolation function (FIF) is a fixed point of the Read–Bajraktarević operator defined on a suitable function space and is constructed via an iterated function system (IFS). In this paper, we considered the generalized affine FIF generated through the IFS defined by the functions $W_n(x,y)=(a_n(x)+e_n,{\alpha }_n(x)y+{\psi }_n(x))$, $n=1,\dots ,N$. We studied the shift of the fractal interpolation curve, by computing the error estimate in response to a small perturbation on ${\alpha }_n(x)$. In addition, we gave a sufficient condition on the perturbed IFS so that it satisfies the continuity condition. As an application, we computed an upper bound of the maximum range of the perturbed FIF.

Fractal interpolation function (FIF) is a new method of constructing new data points within the range of a discrete set of known data points. Consider the iterated functional system defined through the functions $W_n(x,y)=(a_{n}x+e_n,{\alpha }_n(x)y+{\psi }_n(x))$, $n=1,\dots ,N$. Then, we may define the generalized affine FIF $f$ interpolating a given data set $\{(x_n,y_n)\in I\times \mathbb{R},n=0,1,\dots ,N\}$, where $I=[x_0,x_N]$. In this paper, we discuss the smoothness of the FIF $f$. We prove, in particular, that $f$ is $\theta$-hölder function whenever ${\psi }_n$ are. Furthermore, we give the appropriate upper bound of the maximum range of FIF in this case.

The recurrent iterated function systems (RIFS) were first introduced by Barnsley and Demko and generalized the usual iterated function systems (IFS). This new method allowed the construction of more general sets, which do not have to exhibit the strict self similarity of the IFS case and, in particular, the construction of recurrent fractal interpolation functions (RFIF). Given a data set $\{(x_n,y_n)\in I\times \mathbb{R},n=0,1,\dots ,N\}$ where $I=[x_0,x_N]$, we ensured that attractors of RIFS constructed using Geraghty contractions were graphs of some continuous functions which interpolated the given data. Our approach goes beyond the classical framework and provided a wide variety of systems for different approximations problems and, thus, gives more flexibility and applicability of the fractal interpolation method. As an application, we studied the error rates of time series related to the vaccination of COVID-19 using RFIF, and we compared them with the obtained results on the FIF.

In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $f$, we study a new class of fractal interpolation functions which converge to $f$.

Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding significance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure $\mu$ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions defined on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyrière's multifractal Hausdorff and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local Hölder exponents. As an application, we study the relative multifractal binomial measure in symbolic space $\mathrm{\partial }\mathcal{A}$.

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures ${\mathcal{H}}_{\mathit{\mu }}^{q,t}$ and the relative packing measures ${\mathcal{P}}_{\mathit{\mu }}^{q,t}$ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that ${\mathcal{H}}_{\mathit{\mu }}^{q,t}$ and ${\mathcal{P}}_{\mathit{\mu }}^{q,t}$ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.

This paper presents a detailed procedure for deriving a Binet's type formula for the Tribonacci sequence $\{{\mathsf{T}}_n\}$. We examine the limiting distribution of a Markov chain that encapsulates the entire sequence $\{{\mathsf{T}}_n\}$, offering insights into its asymptotic behavior. An approximation of ${\mathsf{T}}_n$ is provided using two distinct probabilistic approaches. Furthermore, we study random sequences of the form $\{{\mathsf{Z}}_0,{\mathsf{Z}}_1,{\mathsf{Z}}_2,{\mathsf{Z}}_n={\mathsf{Z}}_{n-3}+{\mathsf{Z}}_{n-2}+{\mathsf{Z}}_{n-1},n=3,\dots \}$, referred to as the Tribonacci sequence of Random Variables. These sequences, fully defined by their initial random variables, are analyzed in terms of their distributional and limiting properties.

Let $\mu$ and $\nu$ be two compactly supported Borel probability measures on ${\mathbb{R}}^d$ and ${\mathbb{R}}^l$, respectively, and let $q\in \mathbb{R}$ and $h,g$ be two Hausdorff functions. In this paper, we are concerned with evaluation of the lower and upper Hewitt-Stromberg measure of Cartesian product sets, denoted, respectively, by ${\mathsf{H}}_{\mu }^{q,g}$ and ${\mathsf{P}}_{\nu }^{q,h}$, by means of the measure of their components. This is done by the construction of new multifractal measures in a similar manner to Hewitt-Stomberg measures but using the class of all (semi-) half-open binary cubes of covering sets in the definition rather than the class of all balls. Our derived product formula excludes the $0$– $\mathrm{\infty }$ case, and our approach is uniquely applied within an Euclidean space, distinguishing it from those previously utilized in metric spaces. Furthermore, by examining the measures of symmetric generalized Cantor sets, we establish that the exclusion of the $0$– $\mathrm{\infty }$ condition is essential and cannot be omitted.

This paper presents a detailed procedure for determining the probability of return for random walks on $\mathbb{Z}$, whose increment is given by a generalization of a well-known Fibonacci sequence, namely the $k$-Fibonacci-like sequence $(G_{k,n}{)}_n$. Also, we study the size of the set of these walks that return to the origin an infinite number of times, in term of fractal dimension. In addition, we investigate the limiting distribution of an adequate Markov chain that encapsulates the entire Tribonacci sequence $({\mathsf{T}}_n)$ to provide the limiting behavior of this sequence.

In this work, we investigate a class of generalized affine fractal interpolation functions (FIF) with variable parameters, where ordinate scaling is substituted by a real-valued control function. Let $\mathcal{S}$ be an iterated function system (IFS) with the attractor $G_{\mathrm{\Delta }}$, where $\mathrm{\Delta }$ is a given data set. We consider an affine transformation $\omega (\mathrm{\Delta })$ of $\mathrm{\Delta }$, and we define the IFS $\hat{\mathcal{S}}$ with the attractor $G_{\omega (\mathrm{\Delta })}$. We give a sufficient condition so that $G_{\omega (\mathrm{\Delta })}=\omega (G_{\mathrm{\Delta }})$. In addition, we compare the definite integrals of the corresponding FIF and study the additivity property. Some examples will be given, highlighting the effectiveness of our results.