FRACTAL DIMENSION OF MULTIVARIATE α-FRACTAL FUNCTIONS AND APPROXIMATION ASPECTS

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

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 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 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.

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.

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.

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,∞).

Preserving convexity through rational cubic spline fractal interpolation function

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.

Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation ability and data adaptation ability, resulting in unsatisfactory performance. To address these challenges, we introduce the neural basis function by leveraging an untrained neural network as the basis function. Equipped with the proposed neural basis function, we suggest the neural approximation (NeuApprox) paradigm for multivariate function approximation. Specifically, the underlying multivariate function behind the multi-dimensional data is decomposed into a sum of block terms. The clear physically-interpreted block term is the product of expressive neural basis functions and their corresponding learnable coefficients, which allows us to faithfully capture distinct components of the underlying data and also flexibly adapt to new data by readily fine-tuning the neural basis functions. Attributed to the elaborately designed block terms, the suggested NeuApprox enjoys strong approximation ability and flexible data adaptation ability over the hand-crafted basis function-based methods. We also theoretically prove that NeuApprox can approximate any multivariate continuous function to arbitrary accuracy. Extensive experiments on diverse multi-dimensional datasets (including multispectral images, light field data, videos, traffic data, and point cloud data) demonstrate the promising performance of NeuApprox in terms of both approximation capability and adaptability.

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.

A general procedure to define non-smooth versions of classical approximants by means of fractal interpolation functions is proposed. A complete and explicit description in the frequency domain of the functions constructed is obtained through their exact Fourier transforms. In particular, the generalization of the polynomial interpolation is developed. The Lagrange basis of the space of polynomials of degree lower or equal than N is generalized to a basis of fractal polynomials. As a consequence of the process, the density of the polynomial fractal interpolation functions with non-null scale vector in the space of continuous functions in a compact interval is deduced. Furthermore, a method for the interpolation of real data is proposed, by the construction of a fractal function coming from any classical approximant. The convergence of the process when the partition is refined is proved, supposing the convergence of the smooth interpolant.