Energy minimizing associate fractal functions

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.

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 $${\mathcal {C}}[0,1]$$ .

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.

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 alpha - 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.

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.

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 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.

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.

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.

Fractal interpolation functions that share smoothness or nonsmoothness property of the prescribed interpolation data provide a novel method of interpolation. The present paper proposes a new type of rational cubic spline fractal interpolation function which involves two families of free shape parameters and which does not require derivatives at knots for its construction. The scaling factors inherent with the structure facilitate the proposed rational fractal interpolation function to recapture a classical rational cubic spline studied earlier in the literature as a special case. In addition, the scaling factors are the key ingredients that provide fractality to the derivative of the constructed interpolant. Thus, in contrast to the classical nonrecursive rational splines, the proposed rational cubic fractal spline can produce interpolants whose derivatives have irregularity in finite or dense subsets of the interpolation intervals depending on the nature of the problem. Assuming that the original data defining function belongs to the smooth class \(\fancyscript{C}^2\), an upper bound for the interpolation error with respect to the \(L_\infty \)-norm is obtained and the uniform convergence of the rational cubic fractal interpolant is deduced. The developed rational fractal interpolation scheme is illustrated with numerical examples and some possible extensions are exposed.KeywordsFractal SplineFractal Interpolation Functions (FIF)Rational SplineFree Shape ParametersIterated Function Systems (IFS)These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Preserving convexity through rational cubic spline fractal interpolation function

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

Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an
 improved version of iterated function system by using a binary parameter called a signature. The signature
 allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can
 be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously
 differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity,
 and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a
 zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space
 of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are
 studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable
 functions on I for p ∈ [1,∞).