Fractal Polynomial Interpolation

Fractal interpolation, one in the long tradition of those involving the interpolatary theory of functions, is concerned with interpolation of a data set with a function whose graph is a fractal or a self-referential set. The novelty of fractal interpolants lies in their ability to model a data set with either a smooth or a nonsmooth function depending on the problem at hand. To broaden their horizons, some special class of fractal interpolants are introduced and their shape preserving aspects are investigated recently in the literature. In the current article, we provide a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines. To this end, first we shall view polynomial/rational FIFs as α-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of α-fractal function f α incorporates the geometric features such as positivity, monotonicity, and convexity in addition to the regularity inherent in the generating function f . This general theory in conjuction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions. Even though the results obtained in this article are generally enough, we wish to apply it on a specific rational cubic spline with two free shape parameters.

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“…This case is proposed by Barnsley [1] and Navascués [11] as generalization of any continuous function. Here the interpolation data are…”

Section: α-Fractal Functionmentioning

confidence: 99%

Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

Katiyar

Chand

2015

Springer Proceedings in Mathematics &Amp; Statistics

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“…They have been developed both in theory and applications by many authors; see for example [25,40,89,90,95,101,108,109,120]. They provide an alternative view on wavelets, [40,53].…”

Section: Fractal Continuationmentioning

confidence: 99%

Developments in fractal geometry

Barnsley¹,

Vince²

2013

Bull. Math. Sci.

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Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.

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“…These maps tend to bridge the gap between smoothness of classical mathematical objets and pseudo-randomness of experimental variables, breaking in this way their apparent diversity. Navascues and coworkers [17,18,22] contributed to the theory by defining "rough" approximants as perturbation of the functions generally used in classical approximation (polynomial, trigonometric, rational, etc.) via this operator.…”

Section: Prologuementioning

confidence: 99%

“…Following the publication of Fractals Everywhere [2], a beautiful exposition of IFS theory, fractal functions and their applications, various related issues such as calculus, Holder continuity, convergence, stability, smoothness, determination of scaling parameters, and perturbation error have been investigated in the literature [3][4][5][6][7][8][9][10][11][12][13]. The concept of smooth FIFs has been used to generalize the traditional splines [14][15][16][17][18] and to demonstrate that the interaction of classical numerical methods with fractal theory provides new interpolation schemes that supplement the existing ones. Various other extensions of FIFs include multivariable FIFs [7,9,16,[19][20][21][22][23][24][25] generated by using higher-dimensional or recurrent IFSs, the hidden variable FIFs produced by projecting the attractors of vector valued IFSs to a lower-dimensional space.…”

Section: Prologuementioning

confidence: 99%

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

AKB¹

2014

J Appl Computat Math

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This article is targeted to provide a brief and coarse discussion on theory of fractal functions and their recent developments including some of the researches of the authors. Materials on fractal functions provided by this paper is not claimed to be exhaustive, but is intended to be read before or in parallel with technical papers available in the literature on this subject. We have made an earnest attempt to supply an overall flavor of the subject of univarite fractal approximation and useful source of links to assist a novice and perhaps to an expert in fractal functions too.

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Fractal Polynomial Interpolation

Cited by 199 publications

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Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

Developments in fractal geometry

Fractal Functions in Interpolation and Approximation: A Bird's-Eye View

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