Arya Kumar Bedabrata Chand

This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C1-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form pi(θ)∕qi(θ), where pi(θ) and qi(θ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C1-RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

This paper is concerned with interpolation subject to a strip condition on the first order derivative using a class of rational cubic Fractal Interpolation Functions (FIFs). This facilitates the FIF to generate monotonic curves for a given set of monotonic data. The proposed monotonicity preserving rational FIF subsumes and supplements a classical monotonic rational cubic spline. In models leading to the monotonicity preserving interpolation problem wherein the first order derivative of the constructed interpolant is to be nondifferentiable in a finite or dense subset of the interpolation interval, the developed fractal scheme is well-suited in contrast to its classical nonrecursive counterpart. It is shown that the present fractal interpolation scheme has O(h4) accuracy, provided the original function belongs to C4(I) and the parameters involved in the FIF are appropriately chosen.

This paper presents the theory of C1-rational quadratic fractal interpolation surfaces (FISs) over a rectangular grid. First we approximate the original function along the grid lines of interpolation domain by using the univariate C1-rational quadratic fractal interpolation functions (fractal boundary curves). Then we construct the rational quadratic FIS as a blending combination with the x-direction and y-direction fractal boundary curves. The developed rational quadratic FISs are monotonic whenever the corresponding fractal boundary curves are monotonic. We derive the optimal range for the scaling parameters in both positive and negative directions such that the rational quadratic fractal boundary curves are monotonic in nature. The relation between x-direction and y-direction scaling matrices is deduced for symmetric rational quadratic FISs for symmetric surface data. The presence of scaling parameters in the fractal boundary curves helps us to get a wide variety of monotonic/symmetric rational quadratic FISs without altering the given surface data. Numerical examples are provided to demonstrate the comprehensive performance of the rational quadratic FIS in fitting a monotonic/symmetric surface data. The convergence analysis of the monotonic rational quadratic FIS to the original function is reported.

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.

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of C1- rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form (Formula presented.) where pi(x) and qi(x) are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in C2 is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results. © 2013 Springer-Verlag Italia.

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct α-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the α-fractal rational quartic spline when the original function is in 4(I). By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the α-fractal rational quartic spline to 2. The elements of the iterated function system are identified befittingly so that the class of α-fractal function Qα incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ Q. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

We propose a class of affine fractal interpolation surfaces (FISs) that stitch a given set of surface data arranged on a rectangular grid. The proposed FISs are blending of the affine fractal interpolation functions (FIFs) constructed along the grid lines of given interpolation domain. We investigate the stability results of the developed affine FIS with respect to its independent and dependent variables at the grids. These affine FISs preserve the inherited shape of given surface data (like monotonicity, positivity, and convexity), whenever the associated affine FIFs mimic the shape of the univariate data sets along the grid lines of interpolation domain. By using suitable conditions on the scaling factors, we study the monotonicity preserving interpolation via C0-continuous affine FIFs. Under these conditions, apart from one scaling factor, the rest depend only on the functional values but not on both the horizontal contractive factors and slopes at the grids. This weak restriction provides a large flexibility in the selection of the scaling factors for monotonicity preserving C0-continuous affine FIFs/FISs. The positivity criterion for C0-continuous affine FIF is also deduced. © CSP - Cambridge, UK; I&S - Florida, USA, 2014.

This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper α-fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing α-fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper α-fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper α-fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper α-fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper α-fractal functions.

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.