Arya Kumar Bedabrata Chand

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 functions provide a natural deterministic approximation of complex phenomena and also it has self-similarity. Recently, it has been recognized as an internal binary operation, called fractal convolution. In the present article, we obtain Bessel sequences of L2(I× J) composed of product of fractal convolutions, using the identification of L2(I× J) with the tensor product space L2(I) ⊗ L2(J), where I and J are real compact intervals.

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.

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space 𝓛p(I), 1 ≤ p ≤ ∞. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in 𝓛p(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal 𝓛p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in 𝓛p spaces is proven. Further, we derive the fractal analogues of some 𝓛p(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein α-fractal function is developed.

In this paper, we consider some generalization of the Banach contraction principle, namely cyclic contraction and cyclic φ-contraction. For the application to the fractal, we develop new iterated function systems (IFS) consisting of cyclic contractions and cyclic φ-contractions. Further, we discuss about some special properties of the Hutchinson operator associated with the cyclic (c)-comparison IFS.

This article proposes a generalization of the Fourier interpolation formula, where a wider range of the basic trigonometric functions is considered. The extension of the procedure is done in two ways: adding an exponent to the maps involved, and considering a family of fractal functions that contains the standard case. The studied interpolation converges for every continuous function, for a large range of the nodal mappings chosen. The error of interpolation is bounded in two ways: one theorem studies the convergence for Hölder continuous functions and other develops the case of merely continuous maps. The stability of the approximation procedure is proved as well.

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L2(I× J) , the collection of all square integrable functions on the rectangle I× J. The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L2(I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L2(I× J) by the identification of L2(I× J) with the tensor product space L2(I) ⊗ L2(J). In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L2(I× J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way.

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.

Fractal interpolation is more general than the classical piecewise interpolation due to the presence of the scaling factors that describe smooth or non-smooth shape of a fractal curve/surface. We develop the rational cubic fractal interpolation surfaces (FISs) by using the blending functions and rational cubic fractal interpolation functions (FIFs) with two shape parameters in each sub-interval along the grid lines of the interpolation domain. The properties of blending functions and C1-smoothness of rational cubic FIFs render C1-smoothness to our rational cubic FISs. We study the stability aspects of the rational cubic FIS with respect to its independent variables, dependent variable, and first order partial derivatives at the grids. The scaling factors and shape parameters seeded in the rational cubic FIFs are constrained so that these rational cubic FIFs are convex/concave whenever the univariate data sets along the grid lines are convex/concave. For these constrained scaling factors and shape parameters, our rational cubic FIS is convex/concave to given convex/concave surface data.

Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential Lp-functions with a prescribed Lp-function. Further, we apply our construction of fractal functions in Lp-spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in Lp-spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems.