Arya Kumar Bedabrata Chand

This paper introduces a rational Fractal Interpolation Function (FIF), in the sense that it is obtained using a rational cubic spline transformation involving two shape parameters, and investigates its applicability in some constrained interpolation problems. We identify suitable values for the parameters of the corresponding Iterated Function System (IFS) so that it generates positive rational FIFs for a given set of positive data. Further, the problem of identifying the rational IFS parameters so as to ensure that its attractor (graph of the corresponding rational FIF) lies in a specified rectangle is also addressed. With the assumption that the data defining function is continuously differentiable, an upper bound for the interpolation error (with respect to the uniform norm) for the rational FIF is obtained. As a consequence, the uniform convergence of the rational FIF to the original function as the norm of the partition tends to zero is proven.

Coalescence hidden variable fractal interpolation function (CHFIF) proves more versatile than classical interpolant and fractal interpolation function (FIF). Using rational functions and CHFIF, a general construction of A-fractal rational functions is introduced for the first time in the literature. This construction of A-fractal rational function also allows us to insert shape parameters for positivity-preserving univariate interpolation. The convergence analysis of the proposed scheme is established. With suitably chosen numerical examples and graphs, the effectiveness of the positivity-preserving interpolation scheme is illustrated.

This paper investigates some univariate and bivariate constrained interpolation problems using fractal interpolation functions. First, we obtain rational cubic fractal interpolation functions lying above a prescribed straight line. Using a transfinite interpolation via blending functions, we extend the properties of the univariate rational cubic fractal interpolation function to generate surfaces that lie above a plane. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. Uniform convergence of the bivariate fractal interpolant to the original function which generates the data is proven.

The notion of cubic fractal interpolation function (FIF) has received considerable attention in the literature due to its versatility, flexibility and ease of implementation. In this article, we shall view cubic FIFs as a family of C1-continuous fractal functions associated with the traditional C1-continuous cubic spline. General theorems that identify suitable values of the parameters so as to constrain a fractal function and its first derivative within suitable axis-aligned rectangles are reported. By applying these theorems, cubic fractal interpolation of a data set subject to strip conditions on the interpolant and its first derivative is discussed. These results are applied to investigate positivity and monotonicity properties of a hybrid bivariate interpolant over a rectangular region R obtained by blending univariate cubic FIFs via bicubically blended Coons patch. The L∞-norm of the error in approximating a function f ∈ C2(R) with the proposed bivariate interpolant is shown to be of order O(h2) as h → 0.

This paper addresses a method to obtain rational cubic fractal functions, which generate surfaces that lie above a plane via blending functions. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are above the plane whenever the given interpolation data along the grid lines are above the plane. Our rational cubic spline FIS is above the plane whenever the corresponding fractal boundary curves are above the plane. We illustrate our interpolation scheme with some numerical examples.

In this paper, we construct the C1-rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The C1-RCFIF is the fractal design of the traditional rational cubic interpolant of the form pi (θ)/qi (θ), where pi (θ) and qi (θ) are the cubic polynomials with three tension parameters. We derive the uniform error bound between the RCFIF with the original function in C3[x1, xn]. 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.

Inthis article, we propose an interpolation method using a binary parameter called signature such that the graph of the interpolant is an attractor of a suitable zipper rational iterated function system. The presence of scaling factors and signature in the proposed zipper rational quadratic fractal interpolation functions (ZRQFIFs) gives the flexibility to produce a wide variety of interpolants. Using suitable conditions on the scale factor and shape parameter, we construct a C1-continuous ZRQFIF from a C0-continuous ZRQFIF. We also establish the uniform convergence of ZRQFIF to an original data-generating function. Further, we deduce suitable conditions on the IFS parameters and shape parameters to retain the positivity feature associated with a prescribed data by the proposed interpolant.

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 $$\varPhi \in \mathcal {C}^3[x_1, x_n]$$Φ∈C3[x1,xn] with $$h^3$$h3 order of convergence, where $$h$$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.