FRACTAL TRIGONOMETRIC POLYNOMIALS for RESTRICTED RANGE APPROXIMATION
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
Fractal bases for Banach spaces of smooth functions
This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley-Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called -fractal functions. This leads to a bounded linear map on the space which is exploited to prove the existence of a Schauder basis for consisting of smooth fractal functions.
A fractal procedure for monotonicity preserving interpolation
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.
Bivariate shape preserving interpolation: A fractal-classical hybrid approach
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.
A C1-Rational Cubic Fractal Interpolation Function: Convergence and Associated Parameter Identification Problem
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.
Bicubic partially blended rational fractal surface for a constrained interpolation problem
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.
Fractal perturbation preserving fundamental shapes: Bounds on the scale factors
Fractal interpolation function defined through suitable iterated function system provides a method to perturb a function f∈C(I) so as to yield a class of functions fα∈C(I), where α is a free parameter, called scale vector. For suitable values of scale vector α, the fractal functions fα simultaneously interpolate and approximate f. Further, the iterated function system can be selected suitably so that the corresponding fractal function fα shares the quality of smoothness or non-smoothness of f. The objective of the present paper is to choose elements of the iterated function system appropriately in order that fα preserves fundamental shape properties, namely positivity, monotonicity, and convexity in addition to the regularity of f in the given interval. In particular, the scale factors (elements of the scale vector) must be restricted to satisfy two inequalities that provide numerical lower and upper bounds for the multipliers. As a consequence of this process, fractal versions of some elementary theorems in shape preserving interpolation/approximation are obtained. For instance, positive approximation (that is to say, using a positive function) is extended to the fractal case if the factors verify certain inequalities. © 2014 Elsevier Inc.
A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects
The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a L1-cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the Lp-norm, 1≤p≤∞) for the interpolation error of the L1-cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys L2 global smoothness. Consequently, our method offers an alternative to the standard moment construction of L2-cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting L2-cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results. © 2013 Springer Science+Business Media Dordrecht.
Towards a more general type of univariate constrained interpolation with fractal splines
Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420-442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.
On fractal rational functions
This article introduces fractal perturbation of classical rational functions via α-fractal operator and investigates some aspects of this new function class, namely, the class of fractal rational functions. Its specific aims are: (1) to define the fractal rational functions along the lines of the fractal polynomials (2) to extend the Weierstrass theorem of uniform approximation to fractal rational functions (3) to deduce a fractal version of the classical Müntz theorem on rational functions (4) to prove the existence of a Schauder basis for C.I/consisting of fractal rational functions.