Local alpha -fractal interpolation function

Fractal calculus is the calculus involving -integral and -derivative, where in (0,1] is the dimension of the fractals. In defining -integral and -derivative, the mass function and staircase function plays an important role. This chapter discusses the fractal calculus of non-affine fractal interpolation functions namely hidden variable fractal interpolation function and -fractal function. The fractal integral of the hidden variable fractal interpolation function is examined by predefining the initial conditions. Similarly, by predefining the initial conditions and imposing some necessary conditions on the -fractal function, its fractal integral is explored.

Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function $$\varPhi \in \mathcal {C}^3[x_1, x_n]$$ with $$h^3$$ order of convergence, where $$h$$ denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions $$f^{\varvec{\alpha }}\in {\mathcal {C}}(I)$$ , where $$\varvec{\alpha }$$ is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the $$\varvec{\alpha }$$ -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the $$\varvec{\alpha }$$ -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the $$\varvec{\alpha }$$ -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Muntz theorem in $${\mathcal {C}}[0,1]$$ .

Preserving convexity through rational cubic spline fractal interpolation function

In this paper, we explore the concept of dimension preserving approximation of continuous multivariate functions defined on the domain [Formula: see text] (q-times) where [Formula: see text] is a natural number). We establish a few well-known multivariate constrained approximation results in terms of dimension preserving approximants. In particular, we indicate the construction of multivariate dimension preserving approximants using the concept of [Formula: see text]-fractal interpolation functions. We also prove the existence of one-sided approximation of multivariate function using fractal functions. Moreover, we provide an upper bound for the fractal dimension of the graph of the [Formula: see text]-fractal function. Further, we study the approximation aspects of [Formula: see text]-fractal functions and establish the existence of the Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on [Formula: see text] and prove the existence of multivariate fractal polynomials for the approximation.

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.

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 mathcal {L}^2 space associated with the FIF.

The object of this paper is twofold. First, a basic question on fractal function from the perspective of numerical analysis, namely, how to evaluate a Fractal Interpolation Function (FIF) at a prescribed point is addressed by an appeal to a series expansion for FIF. Second is related to the notion of $$\alpha $$ -fractal function, which we call associate fractal function, in the field of fractal approximation. We provide strategies for effective choices of scaling parameters in the associated fractal function. To this end, we minimize some functionals, such as square integral, which represents the total energy or power of a continuous-time signal and Holladay functional, which approximates the total bending energy.

The methods of real data interpolation can be generalized with fractal interpolation. These fractal interpolation functions can be constructed with the so-called iterated function systems. Local iterated function systems are an important generalization of the classical iterated function systems. In order to obtain new approximation methods this methods can be combined with classical interpolation methods. In this paper we focus on the study of the stochastic local fractal interpolation function in the case of a random data set.

Let a data set {(x i,y i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x 0,x N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x i)=y i fori e {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.

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.

In this paper, we demonstrate the existence of a new class of fractal functions in standard function spaces such as the Lebesgue function space and the bounded function space. We also provide the existence of the general version of the so-called ‘ α \alpha -fractal function’. We emphasize the fact that our construction provides some new classes of fractal functions that are not obtained by the previously known classes of α \alpha -fractal functions. We also discuss the stability and the fractal dimension of the newly constructed fractal functions. Lastly, we plot approximate graphs of some newly constructed fractal functions.

In this paper, we investigate the smoothness of non-equidistant fractal interpolation functions. We obtain the Holder exponents of such fractal interpolation functions by using the technique of operator approximation. At last, we discuss the series expressions of these functions and give a Box-counting dimension estimation of “critical” fractal interpolation functions by using our smoothness results.

In this paper, we construct the $$\mathscr {C}^1$$ -rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The $$\mathscr {C}^1$$ -RCFIF is the fractal design of the traditional rational cubic interpolant of the form $$\frac{p_i(\theta )}{q_i(\theta )}$$ , where $$p_i(\theta )$$ and $$q_i(\theta )$$ are the cubic polynomials with three tension parameters. We derive the uniform error bound between the RCFIF with the original function in $$\mathscr {C}^3[x_1,x_n]$$ . When the data set is constrained between two piecewise straight lines, we deduce the sufficient conditions on the parameters of the RCFIF so that it lies between those two lines. Numerical examples are given to support that our method is interactive and smooth.

This chapter provides an overview of several types of fractal interpolation functions that are often studied by many researchers and includes some of the latest research made by the authors. Furthermore, it focuses on the connections between fractal interpolation functions resulting from Banach contractions as well as those resulting from Rakotch contractions. Our aim is to give theoretical and practical significance for the generation of fractal (graph of) functions in two and three dimensions for interpolation purposes that are not necessarily associated with Banach contractions.