Fractal perturbation preserving fundamental shapes: Bounds on the scale factors

Viswanathan, P.; Chand, A. K. B.; Navascués, M. A.

doi:10.1016/j.jmaa.2014.05.019

Cited by 69 publications

(32 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the suitability of FIFs as a bona fide technique for shape preservation has not been fully exploited hitherto. Recently, the authors have initiated shape preserving fractal interpolation and approximation through their researches that appeared in [8][9][10][15][16][17].…”

Section: Prologuementioning

confidence: 99%

A Novel Approach to Surface Interpolation: Marriage of Coons Technique and Univariate Fractal Functions

Chand

Viswanathan

Reddy

2015

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

View full text Add to dashboard Cite

The current article is intended to demonstrate that the theory of fractal functions when applied in conjunction with methods in the classical numerical analysis can supply new solution techniques that supplement and subsume the existing ones. To this end, in the first part of the paper, we review a C 1 -continuous rational cubic fractal interpolation function (FIF) introduced recently [Viswanathan and Chand, Elec. Trans. Numer. Anal. 41 (2014), pp. 420-442]. We carry out the convergence analysis of this univariate rational FIF and determine suitable values of the derivative parameters so that its global smoothness enhances to C 2 . In the subsequent part of the article, we apply Coons technique of transfinite interpolation in order to construct a new kind of C 1 -continuous bivariate fractal interpolation surface.

show abstract

Section: Prologuementioning

confidence: 99%

A Novel Approach to Surface Interpolation: Marriage of Coons Technique and Univariate Fractal Functions

Chand

Viswanathan

Reddy

2015

Springer Proceedings in Mathematics &Amp; Statistics

Self Cite

View full text Add to dashboard Cite

show abstract

“…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%

“…via this operator. Recently [23], the authors investigated conditions on the elements of the IFS so that the perturbation preserves the basic shape properties inherent in the original function, thereby paving a way to shape preserving fractal approximation, an interesting area that needs further exploration at least in our opinion.…”

Section: Prologuementioning

confidence: 99%

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

AKB¹

2014

J Appl Computat Math

View full text Add to dashboard Cite

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.

show abstract

“…The shape properties are mathematically expressed in terms of conditions such as positivity, monotony, and convexity. As a submissive contribution to this goal, Chand and group have initiated the study on shape preserving fractal interpolation and approximation using various families of polynomial and rational IFSs (see, for instance, [5][6][7]20]). These shape preserving fractal interpolation schemes possess the novelty that the interpolants inherit the shape property in question and at the same time the suitable derivatives of these interpolants own irregularity in finite or dense subsets of the interpolation interval.…”

Section: Introductionmentioning

confidence: 99%

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

Katiyar

Chand

2015

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

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.

show abstract

Fractal perturbation preserving fundamental shapes: Bounds on the scale factors

Cited by 69 publications

References 16 publications

A Novel Approach to Surface Interpolation: Marriage of Coons Technique and Univariate Fractal Functions

A Novel Approach to Surface Interpolation: Marriage of Coons Technique and Univariate Fractal Functions

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

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

Contact Info

Product

Resources

About