Search Results (39)

The development of fractal set theory has been strongly driven by the introduction of new classes of fractal sets, among which the attractors of iterated function systems (IFSs) play a central role. In this work, we study a generalization of the classical IFS framework leading to the construction of fractal interpolation functions (FIFs) in which the standard linear ordinate scaling is replaced by a nonlinear contraction. This modification gives rise to a new family of FIFs associated with contractions of the general integral type, offering a flexible and robust approach for the approximation of experimental and irregular data. Furthermore, we introduce a class of generalized iterated function systems defined by mappings acting on product spaces of the form $f:{\mathbb{X}}^m\to \mathbb{X}$, with $m\in {\mathbb{N}}^{*}$. We prove the existence and uniqueness of the corresponding attractor, thereby extending several classical results from the theory of IFS and fractal interpolation. Full article

In this article, the $\alpha$-fractal interpolation function $f^{\alpha }$ corresponding to any function f belonging to the weighted Sobolev space ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is defined. The convergence of sequences of $\alpha$-fractal interpolation functions corresponding to mappings in ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ with respect to the uniform norm as well as the weighted Sobolev norm is discussed. It is proved that the Riemann–Liouville fractional order integral of an $\alpha$-fractal interpolation function of any map $f\in {\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is also a self-referential function interpolating a specific data set. Some aspects of the convergence of the Riemann–Liouville integral of $\alpha$-fractal functions when the original mappings converge are also analyzed. In short, by imposing certain conditions on the base function and the scale vector of a specific iterated function system, fractal perturbations of functions from weighted Sobolev spaces are defined. It is also proved that, under suitable hypotheses, the Riemann–Liouville fractional integral of these fractal mappings on Sobolev spaces is a fractal function of the same kind. Full article

Given a finite set of interpolation data $\{(x_i,y_i)\in I\times \mathbb{R},i=0,1,\dots ,N\}$, $I=[x_0,x_N]$, we construct a class of nonlinear fractal interpolation functions whose graphs are realized as attractors of appropriately defined iterated function systems. In contrast to the classical framework based on uniform contraction mappings, the present approach is built upon an integral-type contraction condition, which extends the standard Banach setting to a more general and flexible context. By applying Branciari’s fixed point theorem, we prove the existence and uniqueness of continuous fractal interpolants associated with these systems. This generalized formulation contains the classical Barnsley fractal interpolation functions as a particular case, while allowing greater adaptability in the modeling of complex and irregular phenomena. As an application, the proposed methodology is implemented on real time-series data describing vaccination dynamics in four different countries, illustrating the effectiveness of the constructed fractal interpolation functions in approximating highly irregular real-world signals. Full article

The reliability of mechanical systems hinges on analyzing the actual surface-to-surface contact area, which critically influences dynamic behavior, friction, material performance, and thermal dissipation. Uneven surfaces lead to incomplete contact, where only a fraction of asperities touch, creating a nominal contact area. This study proposes a novel fractal contact model for various mechanical behaviors between mechanical contact surfaces, integrating the Weierstrass–Mandelbrot fractal function and nonuniform rational B-spline interpolation (NURBS) to model material-dependent actual contact conditions. Furthermore, this research delved into the changes in thermal conductivity across the surfaces of metal materials within a simulated setting. It maintained a contact ratio ranging from 0.038% to 15.2%, a factor that remained unaffected by contact pressure. Both experimental and simulated findings unveiled an actual contact rate spanning from 0.44% to 1.06%, thereby underscoring the distinctive interface behaviors specific to different materials. The proposed approach provides fresh perspectives for investigating material–contact interactions and tackling associated engineering hurdles. Full article

In this paper, we introduce fractal interpolation on complete semi-vector spaces. This approach is motivated by the requirements of the preservation of positivity or monotonicity of functions for some models in approximation and interpolation theory. The setting in complete semi-vector spaces does not requite additional assumptions but is intrinsically built into the framework. For the purposes of this paper, fractal interpolation in the complete semi-vector spaces $C^{+}$ and $L_p^{+}$ is considered. Full article

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov spaces. The spectral theory of families of boundary-interpolating operators is also expanded, offering precise eigenvalue estimates and functional calculus applications. Furthermore, we explore Bessel operators under non-standard measures, such as fractal and weighted geometries, uncovering new analytical phenomena. Key implications include advanced insights into singular integrals, heat kernel behavior, and the boundedness of Riesz transforms, with potential applications in fractal geometry, constrained wave propagation, and mathematical physics. Full article

In this short note, we consider fractal interpolation in the Banach space $V^{\theta }\left(I\right)$ of convex Lipschitz functions defined on a compact interval $I\subset \mathbb{R}$. To this end, we define an appropriate iterated function system and exhibit the associated Read–Bajraktarević operator T. We derive conditions for which T becomes a Ratkotch contraction on a closed subspace of $V^{\theta }\left(I\right)$, thus establishing the existence of fractal functions of class $V^{\theta }\left(I\right)$. An example illustrates the theoretical findings. Full article

This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge to the desired root. Despite this drawback of Steffensen’s method, several researchers have developed higher-order iterative methods based on Steffensen’s scheme. Traub introduced a free parameter in Steffensen’s scheme to obtain the first parametric iteration method, which provides larger basins of attraction for specific values of the parameter. In this paper, we introduce a two-step derivative free fourth-order optimal iteration scheme based on Traub’s method by employing three free parameters and a weight function. We further extend it into a two-step eighth-order iteration scheme by means of memory with the help of suitable approximations of the involved parameters using Newton’s interpolation. The convergence analysis demonstrates that the proposed iteration scheme without memory has an order of convergence of 4, while its memory-based extension achieves an order of convergence of at least $7.993$, attaining the efficiency index $7.{993}^{1/3}\approx 2$. Two special cases of the proposed iteration scheme are also presented. Notably, the proposed methods compete with any optimal j-point method without memory. We affirm the superiority of the proposed iteration schemes in terms of efficiency index, absolute error, computational order of convergence, basins of attraction, and CPU time using comparisons with several existing iterative methods of similar kinds across diverse nonlinear equations. In general, for the comparison of iterative schemes, the basins of iteration are investigated on simple polynomials of the form $z^n-1$ in the complex plane. However, we investigate the stability and regions of convergence of the proposed iteration methods in comparison with some existing methods on a variety of nonlinear equations in terms of fractals of basins of attraction. The proposed iteration schemes generate the basins of attraction in less time with simple fractals and wider regions of convergence, confirming their stability and superiority in comparison with the existing methods. Full article

Following the work on non-stationary fractal interpolation (Mathematics 7, 666 (2019)), we study non-stationary or statistically self-similar fractal interpolation on the Sierpiński gasket (SG). This article provides an upper bound of box dimension of the proposed interpolants in certain spaces under suitable assumption on the corresponding Iterated Function System. Along the way, we also prove that the proposed non-stationary fractal interpolation functions have finite energy. Full article

In this paper, a novel class of rational cubic fractal interpolation function (RCFIF) has been proposed, which is characterized by one shape parameter and a linear denominator. In interpolation for shape preservation, the proposed rational cubic fractal interpolation function provides a simple but effective approach. The nature of shape preservation of the proposed rational cubic fractal interpolation function makes them valuable in the field of data visualization, as it is crucial to maintain the original data shape in data visualization. Furthermore, we discussed the upper bound of error and explored the mathematical framework to ensure the convergence of RCFIF. Shape parameters and scaling factors are constraints to obtain the desired shape-preserving properties. We further generalized the proposed RCFIF by introducing the concept of signature, giving its construction in the form of a zipper-rational cubic fractal interpolation function (ZRCFIF). The positivity conditions for the rational cubic fractal interpolation function and zipper-rational cubic fractal interpolation function are found, which required a detailed analysis of the conditions where constraints on shape parameters and scaling factor lead to the desired shape-preserving properties. In the field of shape preservation, the proposed rational cubic fractal interpolation function and zipper fractal interpolation function both represent significant advancement by offering a strong tool for data visualization. Full article

In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept by introducing the IKC-iterated function/multi-function system using interpolative Kannan operators, including a broader area of mappings. Moreover, we establish the Collage Theorem endowed with the iterated function system (IFS) based on the IKC, and show the well-posedness of the IKC-IFS. Interpolative Kannan contractions are meaningful due to their applications in fractals, offering a more versatile framework for creating intricate geometric structures with potentially fewer constraints compared to classical approaches. Full article

The issue of uneven ground settlement caused by the excavation of subway tunnels represents a significant challenge in the design and construction of subway projects. This paper examined the fractal characteristics of surface settlement caused by subway excavation, employing wavelet transform and fractal theory. Firstly, the noise reduction effects of different wavelet functions, decomposition levels, threshold functions, and threshold selection rules were evaluated using the SNR and RMSE. Subsequently, 291 data points were derived from 18 interpolation points through fractal interpolation, representing a utilization of only 18% of the original data, to enhance the original monitoring data information by a factor of 2.94. The average relative error between the fractal interpolation results and the monitoring data was approximately 13%, which was indicative of a high degree of accuracy. Finally, the fractal dimension of the monitoring curves under different monitoring frequencies was calculated using the box-counting method. The denoising effect of the dbN wavelet function was found to be superior to that of the symN wavelet function in the denoising process of subway construction surface deformation monitoring data. Furthermore, the hard threshold method was observed to be more effective than the soft threshold method. As the monitoring frequency decreased, the fractal dimension of the deformation curves showed an overall decreasing trend. This indicated that high-frequency monitoring could capture more details and complexity of the surface settlement, while low-frequency monitoring led to a gradual flattening of the curves and a decrease in details. Full article

Time series of financial data are both frequent and important in everyday practice. Numerous applications are based, for example, on time series of asset prices or market indices. In this article, the application of fractal interpolation functions in modelling financial time series is examined. Our motivation stems from the fact that financial time series often present fluctuations or abrupt changes which the fractal interpolants can inherently model. The results indicate that the use of fractal interpolation in financial applications is promising. Full article

In this article, we consider an iterated functions system on the non-Euclidean real projective plane which has a linear structure. Then, we study the fractal dimension of the associated curve as a subset of the projective space and like a set of the Euclidean space. At the end, we initiate a dual real projective iterated function system and pose an open problem. Full article

The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented. Full article