Hidden Variable Fractal Interpolation Functions Research Articles

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.

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.

On the Variable Order Fractional Calculus Characterization for the Hidden Variable Fractal Interpolation Function

In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden variable fractal interpolation function. Moreover, the conditions for the variable fractional order μ on a specified range are also derived. It is observed that, under certain conditions, the Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus of the hidden variable fractal interpolation function are again the hidden variable fractal interpolation functions interpolating the new data set.

ANALYSIS ON WEYL–MARCHAUD FRACTIONAL DERIVATIVE FOR TYPES OF FRACTAL INTERPOLATION FUNCTION WITH FRACTAL DIMENSION

In this paper, the Weyl–Marchaud fractional derivative of various fractal interpolation functions (FIFs) like linear FIF, quadratic FIF, hidden variable FIF and [Formula: see text]-FIF is investigated. Further, the fractal dimension of the quadratic FIF is estimated and it is compared with the order of the Weyl–Marchaud fractional derivative. Besides, this paper shows that the Weyl–Marchaud fractional derivative of all FIFs is again FIFs if the order of the fractional derivative meets the necessary condition.

Riemann Liouville fractional integral of hidden variable fractal interpolation function

Abstract In this paper, we study Riemann Liouville fractional integral of hidden variable fractal interpolation function (HVFIF) constructed by functions whose Lipschitz exponents are in (0, 1]. Firstly, we present a construction of HVFIF using functions of which Lipschitz exponents are in (0, 1], so that the Riemann Liouville fractional integral of the HVFIF becomes a fractal interpolation function, and give an example where Lipschitz exponents of functions of IFS are in (0, 1]. Secondly, we prove that the Riemann Liouville fractional integral is also a HVFIF with function vertical scaling factors defined newly. Finally, we give the graphs of 0.8- and 0.2-order fractional integrals of the HVFIFs constructed in the above example.

Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors

We estimate the bounds of box-counting dimension of hidden variable fractal interpolation functions (HVFIFs) and hidden variable bivariate fractal interpolation functions (HVBFIFs) with four function contractivity factors and present analytic properties of HVFIFs which are constructed to ensure more flexibility and diversity in modeling natural phenomena. Firstly, we construct the HVFIFs and analyze their smoothness and stability. Secondly, we obtain the lower and upper bounds of box-counting dimension of the HVFIFs. Finally, in the similar way, we get the lower and upper bounds of box-counting dimension of HVBFIFs constructed in [21].

HIDDEN VARIABLE RECURRENT FRACTAL INTERPOLATION FUNCTIONS WITH FUNCTION CONTRACTIVITY FACTORS

In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIFs) with four function contractivity factors. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed using a recurrent iterated function system (RIFS). In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation functions. RIFSs produce fractal sets with local self-similarity structure. Therefore, the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). The HVFIF is neither self-similar nor self-affine one. Hence, the HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The contractivity factors of IFS are very important one that determines characteristics of FIFs. The IFS and RIFS with function contractivity factors can describe the fractal objects in nature better than one with constant contractivity factors. To ensure higher flexibility and diversity of the construction of the FIFs, we present constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

CONSTRUCTION OF NONLINEAR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR STABILITY

This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.

Reproducing Kernel Hilbert Space and Coalescence Hidden-variable Fractal Interpolation Functions

Abstract Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are important tools which have been found to be incredibly useful in many areas like machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. In the present paper, the space of Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is demonstrated to be an RKHS and its associated kernel is derived. This extends the possibility of using this new kernel function, which is partly self-affine and partly non-self-affine, in diverse fields wherein the structure is not always self-affine.

Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions

Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.

Approximation using hidden variable fractal interpolation function

The notion of hidden variable fractal interpolation provides a method to approximate functions that are self-referential or non-self-referential, and consequently allows great flexibility and diversity for the fractal modeling problem. The current article intends to apply hidden variable fractal interpolation to associate a class of $\mathbb R^2$-valued continuous fractal functions with a prescribed continuous function. Suitable values of the parameters are identified so that the fractal functions retain positivity and regularity of the germ function. As an application of the developed theory, we obtain positive $\mathcal C^1$-cubic spline hidden variable fractal interpolation functions corresponding to a prescribed set of positive data, thus initiating a new approach to shape preserving approximation via hidden variable fractal function. Depending on the values of the parameters, these positive interpolants can reflect the self-referentiality or non-self-referentiality of the original data defining function and fractality of its derivative. Therefore, the present scheme outperforms the traditional nonrecursive positivity preserving $\mathcal C^1$-cubic spline interpolation scheme and its fractal extension studied recently in the literature.

NATURAL CUBIC SPLINE COALESCENCE HIDDEN VARIABLE FRACTAL INTERPOLATION SURFACES

Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid [Formula: see text] through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function [Formula: see text], and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

SENSITIVITY ANALYSIS FOR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR MOMENTS

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

Stability of affine coalescence hidden variable fractal interpolation functions

The stability of an affine coalescence hidden variable fractal interpolation function is proved in a general set up in the present work, by establishing that any small perturbation in the generalized interpolation data leads to a small perturbation in the corresponding affine coalescence hidden variable fractal interpolation function.

CUBIC SPLINE COALESCENCE FRACTAL INTERPOLATION THROUGH MOMENTS

This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

Spline coalescence hidden variable fractal interpolation functions

This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (1989), when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.