Fractal Interpolation Functions Research Articles

Generalized G-Hausdorff space and applications in fractals

In this paper, we introduce the concept of a G-Hausdorff space and show how the results established in the usual metric space can be generalized to the G-metric space. The proven results are used to propose an iterated function system (IFS) called G-IFS. Additionally, the existence of the fractal associated with this construction is demonstrated. This paper shows how non-affine transformations and fractal interpolation functions (FIFs) can be used to approximate fractals by G-IFS. This paper contributes to the understanding of fractal geometry and its applications in mathematics and other fields.

Bivariate Fractal Interpolation Functions on Triangular Domain for Numerical Integration and Approximation

The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided such that the actual double integral coincides with the integral value calculated using fractal theory. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Adequate examples are presented to illustrate the method of construction and to verify the value of double integration.

Remarks on the integral transform of non-linear fractal interpolation functions

This paper examines the integral transform of fractal interpolation functions with function scaling factors. Initially, the integral transform of quadratic fractal interpolation function, quadratic hidden variable fractal interpolation function and α-fractal functions with function scaling factors is investigated. Using a specific application of kernel functions, we further explore the integral transform of fractal functions such as the Laplace transform and the Laplace–Carson transform.

Financial Time Series Modelling Using Fractal Interpolation Functions

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.

Approximation by the linear fractal interpolation functions with the same fractal dimension

Approximation by the linear fractal interpolation functions with the same fractal dimension

Fractal dimension analysis of stock prices of selected resulting companies after mergers and acquisitions.

This study focuses on the short-run wealth of listed firms' shareholders. Currently, all of the resulting organizations offer competitive pricing tactics to create a superior environment for our ongoing establishment. Some time ago, it was noted that a merger occurred, although some functions and technology integration remained with the previous structure. In this paper, it has been discovered that merger and acquisition deals have an impact on the firm's value; in other words, we can view it as shareholders' wealth or unit depending on the stock price after the announcement of merger and acquisition deals in the short term. Furthermore, we focused on influencing variables on stock prices after the announcement of merger and acquisition transactions, which is measured as a percentage change in the stock prices of the listed resulting firms. Furthermore, this research is based on secondary data sources from reputable organizations. It primarily uses the NSE database and website to evaluate announcements and stock prices of the twenty-nine publicly traded companies. Markets respond to investors' emotions and market expertise. When acquirers have a strong market position, market capitalization rises in other segments. However, it is declining due to a lack of supportive finances. To determine the impact of merger and acquisition announcement deals on stock price changes, average abnormal return and cumulative average abnormal return with the capital asset pricing model (CAPM) (CAPM reaction to changes) were used to identify the acquiring company's stock price reaction. We investigated its impact on the fluctuation of share prices posted on stock exchanges using fractal interpolation functions. This is due to greater investment by acquirer businesses in target companies as well as investor expectations for specific stock market strongholds.

Local -fractal interpolation function.

Constructions of the (global) fractal interpolation functions on standard function spaces got a lot of attention in the last centuries. Motivated by the newly introduced local fractal functions corresponding to a local iterated functions system which is the generalization of the traditional iterated functions system we construct the local non-affine - fractal functions in this article. A few examples of the graphs of these functions are provided. A fractal operator which takes the classical function to its local fractal counterpart is defined and some of its properties are also studied.

Propagation of ion-acoustic wave and its fractal representations in spin polarized electron plasma

Propagation of small-amplitude quantum ion-acoustic waves and its fractal representations is investigated in an electron-ion quantum plasma with separated spin electrons in the framework of the KdV and EKdV equations derived using reductive perturbation technique. These two equations are transformed into planar dynamical systems by using suitable traveling wave transformation. Qualitatively different phase portraits of these two systems are drawn with their respective Sagdeev’s pseudopotential curves and surface plots. Distinct orbits in the phase portraits give rise to distinct wave solutions. Periodic and superperiodic wave solutions are investigated numerically and solitary wave solutions are derived analytically. The effect of parameters such as Mach number, direction cosine, spin polarization and frequency ratio on these wave solutions are presented. For instance, it is seen that the spin density polarization ratio has an impressive effect on the amplitude of the wave solution, while the frequency ratio has no effect on its amplitude. Also, we have provided a physical explanation for our finding. Further, the dynamical features of the original system and reconstructed system using fractal interpolation function are investigated under the external forcing term. Chaotic and quasiperiodic phenomena are observed for different initial conditions. The results may help to better understand the degenerate electron gas that exists in dense astrophysical bodies.

Nonlinear fractal interpolation functions on the Koch curve

Nonlinear fractal interpolation functions defined on the Koch curve by replacing the usual Banach contractions used to define (linear) fractal interpolation functions with Rakotch contractions are considered. This allows to take care of the nonlinear term in the associated Read–Bajraktarević operator, respectively, the underlying iterated function system. Moreover, we present an extremely explicit example to demonstrate the effectiveness of the obtained results.

Reproducing Kernel Hilbert Spaces of Smooth Fractal Interpolation Functions

The theory of reproducing kernel Hilbert spaces (RKHSs) has been developed into a powerful tool in mathematics and has lots of applications in many fields, especially in kernel machine learning. Fractal theory provides new technologies for making complicated curves and fitting experimental data. Recently, combinations of fractal interpolation functions (FIFs) and methods of curve estimations have attracted the attention of researchers. We are interested in the study of connections between FIFs and RKHSs. The aim is to develop the concept of smooth fractal-type reproducing kernels and RKHSs of smooth FIFs. In this paper, a linear space of smooth FIFs is considered. A condition for a given finite set of smooth FIFs to be linearly independent is established. For such a given set, we build a fractal-type positive semi-definite kernel and show that the span of these linearly independent smooth FIFs is the corresponding RKHS. The nth derivatives of these FIFs are investigated, and properties of related positive semi-definite kernels and the corresponding RKHS are studied. We also introduce subspaces of these RKHS which are important in curve-fitting applications.

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.

On the variable order fractional calculus of fractal interpolation functions

In this paper, a new idea of exploring the variable order fractional calculus on the fractal interpolation functions is addressed. The studies of constant order fractional calculus on the fractal functions are generalized to the case of variable order. The Riemann–Liouville variable order fractional integral (/derivative) and the Weyl–Marchaud variable order fractional derivative are investigated over the different kinds of fractal interpolation functions (FIFs). Furthermore, the necessary conditions of variable fractional order p(z) defined on the interval $$[x_0,x_N]$$ are also derived. It is observed that under the derived conditions, both the Riemann–Liouville variable order fractional integral (/derivative) and the Weyl–Marchaud variable order fractional derivative of FIFs are again FIFs interpolating new data set.

Analytical and dimensional properties of fractal interpolation functions on the Sierpiński gasket

In this article, we construct the fractal interpolation functions (FIFs) on the Sierpiński gasket (SG) by taking data set at the nth level. We discuss the oscillation space, $$\mathcal {L}_q$$ space and some other function spaces on SG, and determine some conditions under which, this FIF belongs to these spaces. We obtain the fractal dimension of the graph of the FIF and the Hausdorff dimension of the invariant measure supported on the graph of the FIF. We also prove that this FIF has finite energy. After that, we discuss the restrictions of the FIF on the bottom line segment of SG and prove that the restriction is again an FIF on the bottom line. We estimate the fractal dimension of the graph of the Riemann-Liouville fractional integral of the restrictions of the FIF on the bottom line segment of SG. Lastly, we prove that the Hausdorff and box-counting dimension of the graph of the restriction of any harmonic function on the bottom line segment of SG are 1 and for the Riemann- Liouville fractional integral of this restriction function, these dimensions are also 1.

Monotone Data Modelling Using Rational Cubic Fractal Interpolation Function

Geometric modelling of several intricate and complex structures such as trees, mountains, clouds, ferns, geographic topography, and coastlines is challenging in computer graphics. Traditional splines such astrigonometric, polynomial, exponential, and rational fail to simulate this significant class of complex structures, which are highly irregular in nature. For this purpose, this research develops a novel cutting-edgemethod for synthesizing and modelling structures. The proposed technique; C 1 fractal interpolation function (FIF) builds an iterated function system (IFS) by integrating fractal calculus and rational cubic polynomial functions. Appropriate conditions on scaling and shape parameters are derived to help maintain the inherited shape qualities of the data. Experiments in numerous scientific domains, such as the pharmaceutical and chemical industries have been presented as an example, to confirm the usefulness of the suggested model. Moreover, the graphic results demonstrated that the developed monotone hybrid model (MHM) offers a heterogeneous method for gathering data with a monotone structure.

Construction and box dimension of the composite fractal interpolation function

In this paper, we construct the composite fractal interpolation function and prove that the compositeness of two general fractal interpolation functions is still a fractal interpolation function. And its generating iterated function system is obtained from given iterated function systems. Finally, the box dimension of the composite fractal interpolation function is discussed. We give an inequality estimate of the upper and lower box dimensions of the composite fractal interpolation function.

Rational quadratic trigonometric spline fractal interpolation functions with variable scalings

Rational quadratic trigonometric spline fractal interpolation functions with variable scalings

Dimensional study of COVID-19 via fractal functions.

The present paper deals with the modeling of the COVID-19 via fractal interpolation function(FIF) and the estimation of the dimension of constructed FIF. Further, we determine the adjoint of the fractal operator defined on space associated with theFIF.

Explicit relation between Fourier transform and fractal dimension of fractal interpolation functions

Explicit relation between Fourier transform and fractal dimension of fractal interpolation functions

Histopolating fractal functions

Fractal interpolation functions that produce smooth and non-smooth approximants constitute an advancement to various classical nonrecursive methods of approximation. However, in fractal approximation, the emphasis so far has been on the investigation of continuous functions interpolating prescribed data sets. In this article, we exclude the two properties – continuity and interpolation – inherent in fractal interpolation, and introduce the concept of fractal histopolation. That is, we study the area-matching properties of fractal functions that are integrable but not necessarily continuous.

ON THE CLASSICAL INTEGRAL OF FRACTAL FUNCTIONS

In this paper, the integral of classical fractal interpolation function (FIF) and A-fractal function is explored for both the cases of constant and variable scaling factors. The definite integral for the classical FIF in the closed interval of [Formula: see text] is estimated. The novel notion of affine-quadratic FIF is introduced and integrated for both constant and variable scaling factors. It is demonstrated that its integral is not an affine-quadratic FIF, however it is a FIF. Similarly, by choosing the vertical scaling factors as constants and variables, A-fractal function is integrated. Further, by assuming certain condition on the block matrix, it is shown that like the original A-fractal function its integral is also an attractor for the iterated function system.