Arya Kumar Bedabrata Chand

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.

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.