Fractal Interpolation Functions Research Articles

Multifractal analysis of fractal interpolation functions

This paper presents a novel algorithm to utilize multifractal spectrum as a quantitative measure for the fractal interpolation functions with respect to scaling factor and fractional order. As of yet, there were no error estimation techniques to interpret the fractal interpolation functions in the literature. To bridge this gap, this paper sketches multifractality as a quantitative measure for inquiring and comparing the effects of different scaling factors. The proposed algorithm for analyzing the multifractal measure depends on the probability measure of data points, which fractal function passes through, enabling to effectively discuss the heterogeneity of fractal interpolation functions. In addition, the impact of fractional orders on the fractional derivative (integral) of fractal interpolation functions is also discussed tailoring the multifractal measure.

Positivity-Preserving Rational Cubic Fractal Interpolation Function Together with Its Zipper Form

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.

On the box dimension of recurrent fractal interpolation functions defined with Matkowski contractions

On the box dimension of recurrent fractal interpolation functions defined with Matkowski contractions

Analytical properties and the box-counting dimension of nonlinear hidden variable recurrent fractal interpolation functions constructed by using Rakotch's fixed point theorem

Rakotch contraction is a generalization of Banach contraction, which implies that in the case of using Rakotch's fixed point theorem, we can model more objects than using Banach's fixed point theorem. Moreover, hidden variable recurrent fractal interpolation functions (HVRFIFs) with Hölder function factors are more general than the fractal interpolation functions (FIFs), recurrent FIFs and hidden variable FIFs with Lipschitz function factors. We demonstrate that HVRFIFs can be constructed using the Rakotch's fixed point theorem, and then investigate the analytical and geometric properties of those HVRFIFs. Firstly, we construct a nonlinear hidden variable recurrent fractal interpolation functions with Hölder function factors on the basis of a given data set using Rakotch contractions. Next, we analyze the smoothness of the HVRFIFs and show that they are stable on the small perturbations of the given data. Finally, we get the lower and upper bounds for their box-counting dimensions.

Zipper rational fractal interpolation functions

Zipper rational fractal interpolation functions

Contractive Multivariate Zipper Fractal Interpolation Functions

In this paper we introduce a new general multivariate fractal interpolation scheme using elements of the zipper methodology. Under the assumption that the corresponding Read-Bajraktarevic operator is well-defined, we enlarge the previous frameworks occurring in the literature, considering the constitutive functions of the iterated function system whose attractor is the graph of the interpolant to be just contractive in the last variable (so, in particular, they can be Banach contractions, Matkowski contractions, or Meir-Keeler contractions in the last variable). The main difficulty that should be overcome in this multivariate framework is the well definedness of the above mentioned operator. We provide three instances when it is guaranteed. We also display some examples that emphasize the generality of our scheme.

A novel fractal interpolation function algorithm for fractal dimension estimation and coastline geometry reconstruction: a case study of the coastline of Kingdom of Saudi Arabia

A novel fractal interpolation function algorithm for fractal dimension estimation and coastline geometry reconstruction: a case study of the coastline of Kingdom of Saudi Arabia

Fractal Interpolation Function based Thin Wire Antennas

This paper presents an approach for the design of wire antennas based on fractal interpolation functions (FIFs). The interpolation points and the contraction factors of the FIFs are chosen as free parameters to modify the antenna geometry. The proposed structures’ gain and radiation pattern can be optimized using FIF parameters. Producible prefractal antennas obtained in the intermediate iterations of fractal generation have compact sizes compared to classical counterparts. The error in prefractal geometry and the original fractal is bounded, and can be determined in terms of the finest producible detail’s dimensions. The emerging structures have multiband behavior due to their self-similar and symmetric nature. To illustrate the approach, we have provided finite element based simulations for several prefractal antennas. |S11|, the gain, the radiation efficiency, the radiation patterns, and feed point impedances for the demonstrated antennas are calculated numerically. The results indicate that produced antennas can be used in applications that require limited mechanical size, multiple operating bands, and controlled radiation patterns.

Some results on the space of rational cubic fractal interpolation functions

Some results on the space of rational cubic fractal interpolation functions

A very general framework for fractal interpolation functions

M. Barnsley introduced the concept of fractal interpolation function as an alternative method for construction of interpolation functions. Such a function is the attractor of an iterated function system comprising Banach contractions (on the product of two real compact intervals) with respect to the second variable. In the present paper we enlarge classical Barnsley's framework by allowing the constitutive functions of the system to be Edelstein contractions in the second variable. As the class of Edelstein contractions contains the class of Matkowski contractions, the class of Meir-Keeler contractions, the class of F-contractions and the class of θ-contractions, we provide a much more flexible framework for the construction of fractal interpolation functions. We also obtain a result concerning the estimation of lower and upper box dimensions of the graph of a fractal interpolation function constructed via our general method.

On the stability of Fractal interpolation functions with variable parameters

<abstract><p>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) = \big(a_n(x)+e_n, \alpha_n(x) y +\psi_n(x)\big) $, $ n = 1, \ldots, 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.</p></abstract>

CONSTRUCTION OF MONOTONOUS APPROXIMATION BY FRACTAL INTERPOLATION FUNCTIONS AND FRACTAL DIMENSIONS

In this paper, we research on the dimension preserving monotonous approximation by using fractal interpolation techniques. A constructive result of the approximating sequence of self-affine continuous functions has been given, which can converge to the object continuous function of bounded variation on [Formula: see text] monotonously and unanimously, meanwhile their graphs can be any value of the Hausdorff and the Box dimension between one and two. Further, such approximation for continuous functions of unbounded variation or even general continuous functions with non-integer fractal dimension has also been discussed elementarily.

Zipper Quintic Fractal Interpolation Function for Curve Fitting

Zipper Quintic Fractal Interpolation Function for Curve Fitting

FRACTAL SURFACES INVOLVING RAKOTCH CONTRACTION FOR COUNTABLE DATA SETS

In this paper, we prove the existence of the bivariate fractal interpolation function using the Rakotch contraction theory and iterated function system for a countable data set. We also give the existence of the invariant Borel probability measure supported on the graph of the bivariate fractal interpolation function. In particular, we highlight that our theory encompasses the bivariate fractal interpolation theory in both finite and countably infinite settings available in literature.

A traffic-fractal-element-based congestion model considering the uneven distribution of road traffic

The uneven distribution of road traffic has been proven to play a crucial role in congestion evolution within urban transportation networks. Establishing congestion models to explore congestion evolution under its impact is of great significance for predicting congestion. Current congestion models commonly describe the uneven distribution of traffic on roads through continuous differential equations or the simulation of vehicle operational rules by which to model traffic flow dynamics. However, these models fail to directly reveal the mechanism by which the uneven distribution characteristics affect congestion, which is not conducive to uncovering the fundamental factors that affect congestion evolution and general laws for predicting congestion. Therefore, this paper first constructs traffic fractal elements (TFE) based on the fractal characteristics exhibited by uneven traffic distribution. A local parameter namely the load duty ratio (LDR) is introduced to represent the uneven distribution level of TFE. By employing the fractal interpolation function (FIF), we model road traffic as the iterations of TFE. Then we define the fractal evolution rule and the load transfer rule to inscribe the influence relationship between LDR of TFE and global congestion, ultimately establishing a TFE-based congestion model. Furthermore, the proposed TFE-based model is validated using empirical data from AMAP and simulation data from a calibrated Cellular Automata (CA) model. Finally, by comparing the results with a CA-based congestion model, the LDR of TFE is confirmed to be a comprehensive and indicative parameter affecting congestion evolution. By simulating and fitting the congestion evolution data under different LDRs, our proposed model demonstrates its value in congestion prediction and analysis.

Improve Fuzzy Inventory Model of Fractal Interpolation with Vertical Scaling Factor

The inventory model is used to determine the optimal inventory of a product. In certain cases, parameters in the inventory model are uncertain. Fractal interpolation techniques can be used to overcome parameter with uncertainty. Fractal interpolation results are affected by the fractal interpolation function and the vertical scaling factor. The vertical scaling factor is positive and less than 1. In this study, fractal interpolation techniques are introduced with variations in vertical scaling factor to overcome the uncertainty of demand data in inventory models. Furthermore, the interpolation results are used in fuzzy inventory models and expressed by Trapezoidal Fuzzy Number. This paper considers an inventory model with varying demand to optimize rice inventory. Based on the data obtained, the accuracy level will increase for the vertical scaling factor values close to 1. Optimal rice inventory of each successive fuzzy parameter is 1447963, 1013914, 504950, 215312. If the cost parameter is increased, then the amount of inventory is decreases.

Smoothness analysis and approximation aspects of non-stationary bivariate fractal functions

The present note aims to establish the notion of non-stationary bivariate α-fractal functions and discusses some of their approximation properties. We see that using a sequence of iterated function systems generalizes the existing stationary fractal interpolation function (FIF). Also, we show the existence of Borel probability fractal measures supported on the graph of the non-stationary fractal function. Further, we define a fractal operator associated with the constructed non-stationary FIFs, and many applications of this operator such as fractal approximation and the existence of fractal Schauder bases are observed. In the end, we study the constrained approximation with the proposed interpolant.

Applications of fractal interpolants in kernel regression estimations

Generating a function from its finite set of samples is widely studied in data fitting problems. The theory of nonparametric modeling has been developed by many researchers, and several types of estimators have been established in the literature. On the other hand, the fractal theory provides new technologies for making complicated curves and fitting experimental data, and fractal interpolation functions are established as generalizations of traditional interpolation techniques. The aim of this paper is to investigate applications of (smooth) fractal interpolation functions in nonparametric kernel regressions. We discuss a key sufficient condition in detail in the construction of (smooth) fractal interpolation functions. Then the Priestley–Chao estimator is considered, and we establish its fractal perturbation. The raw data we consider in our example is the Crude Oil WTI Futures daily highest price from 2021/10/13 to 2022/7/29. We take 11 samples from the raw data to construct a fractal perturbation of the Priestley–Chao regression function. SLSQP is applied to find the optimal values of hyperparameters so that the given empirical mean squared error is minimum. We see that nonparametric kernel regressions for finite sample data are widely used in applications, and we can decrease the empirical mean squared error further by considering fractal perturbations of these regression functions.

On the integral transform of fractal interpolation functions

This paper explores the integral transform of two distinct fractal interpolation functions, namely the linear fractal interpolation function and the hidden variable fractal interpolation function with variable scaling factors. Further, with a particular application of kernel functions, we investigate the integral transform of fractal functions, such as the Laplace transform and the Laplace Carson transform. Moreover, we show that the compositeness of two fractal interpolation functions, f1 in {tɛ,xɛ} and f2 in {xɛ,zɛ} remains a fractal interpolation function. It also generates iterated function system from given iterated function systems. In addition to this, the study is carried out on the composite linear fractal interpolation function of the integral transform, the Laplace transform, and the Laplace Carson transform.

Box dimension of generalized affine fractal interpolation functions

Let f be a generalized affine fractal interpolation function with a vertical scaling function S . In this paper, we study \mathrm{dim}_B \Gamma f , the box dimension of the graph of f , under the assumption that S is a Lipschitz function. By introducing vertical scaling matrices, we estimate the upper and the lower bounds of oscillations of f . As a result, we obtain an explicit formula of \mathrm{dim}_B \Gamma f under certain constraint conditions.