Arya Kumar Bedabrata Chand

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.

Fractal interpolation functions (FIFs) developed through iterated function systems prove more general than their classical counterparts. However, the theory of fractal interpolation functions in the domain of shape preserving interpolation is not fully explored. In this paper, we introduce a new kind of iterated function system (IFS) involving rational functions of the form pn.x/ qn.x/, where pn.x/ are quadratic polynomials determined through the interpolation conditions of the corresponding FIF and qn.x/ are preassigned quadratic polynomials involving one free shape parameter. The presence of the scaling factors in our rational FIF adds a layer of flexibility to its classical counterpart and provides fractality in the derivative of the interpolant. The uniform convergence of the rational quadratic FIF to the original data generating function is established. Suitable conditions on the rational IFS parameters are developed so that the corresponding rational quadratic fractal interpolant inherits the positivity property of the given data.

IFS constitutes one of the powerful tools to generate fractal sets. Recently, a cyclic map is used in IFS to construct a new class of fractals. This paper is an effort to study multivalued IFSs with various types of cyclic multivalued maps such as cyclic multivalued ϕ -contraction, cyclic multivalued Meir–Keeler contraction and cyclic multivalued contractive which are generalizations of contraction map, and the construction of fractals with the help of these IFSs have been established.

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.