Distribution of Noise in Linear Recurrent Fractal Interpolation Functions for Data Sets with α -Stable Noise
In this study, we construct a linear recurrent fractal interpolation function (RFIF) with variable scaling parameters for data set with α -stable noise (a generalization of Gaussian noise) on its ordinate, which captures the uncertainty at any missing or unknown intermediate point. The propagation of uncertainty in this linear RFIF is investigated, and a method for estimating parameters of the uncertainty at any interpolated value is provided. Moreover, a simulation study to visualize uncertainty for interpolated values is presented.
Fractal approximants on the circle
A methodology based on fractal interpolation functions is used in this work to define new real maps on the circle generalizing the classical ones. The power of fractal methodology allows us to generalize any other interpolant, both smooth and non-smooth, but the important fact is that this technique provides one of the few methods of non-differentiable interpolation. In this way, it constitutes a func- tional model for chaotic processes. In this article we study a generalization of some approximation formulae proposed by Dunham Jackson both in classical and fractal cases.
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.
Constrained 2D data interpolation using rational cubic fractal functions
In this paper, we construct the C1-rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The C1-RCFIF is the fractal design of the traditional rational cubic interpolant of the form pi (θ)/qi (θ), where pi (θ) and qi (θ) are the cubic polynomials with three tension parameters. We derive the uniform error bound between the RCFIF with the original function in C3[x1, xn]. When the data set is constrained between two piecewise straight lines, we deduce the sufficient conditions on the parameters of the RCFIF so that it lies between those two lines. Numerical examples are given to support that our method is interactive and smooth.
Toward a unified methodology for fractal extension of various shape preserving spline interpolants
Fractal interpolation, one in the long tradition of those involving the interpolatary theory of functions, is concerned with interpolation of a data set with a function whose graph is a fractal or a self-referential set. The novelty of fractal interpolants lies in their ability to model a data set with either a smooth or a nonsmooth function depending on the problem at hand. To broaden their horizons, some special class of fractal interpolants are introduced and their shape preserving aspects are investigated recently in the literature. In the current article, we provide a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines. To this end, first we shall view polynomial/rational FIFs as α-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of α-fractal function fα incorporates the geometric features such as positivity, monotonicity, and convexity in addition to the regularity inherent in the generating function f. This general theory in conjuction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions. Even though the results obtained in this article are generally enough, we wish to apply it on a specific rational cubic spline with two free shape parameters.
Positivity Preserving Rational Quartic Spline Zipper Fractal Interpolation Functions
In this paper, we introduce a class of novel C1 -rational quartic spline zipper fractal interpolation functions (RQS ZFIFs) with variable scalings, where rational spline has a quartic polynomial in the numerator and a cubic polynomial in the denominator with two shape control parameters. We derive an upper bound for the uniform error of the proposed interpolant with a C3 data generating function, and it is shown that our fractal interpolant has O(h2) convergence and can be increased to O(h3) under certain conditions. We restrict the scaling functions and shape control parameters so that the proposed RQS ZFIF is positive, when the given data set is positive. Using this sufficient condition, some numerical examples of positive RQS ZFIFs are presented to support our theory.
Positivity preserving rational cubic trigonometric fractal interpolation functions
In this paper, we propose a family of (Formula presented.)-rational cubic trigonometric fractal interpolation function (RCTFIF) to preserve positivity inherent in a set of data. The proposed RCTFIF is a generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form (Formula presented.), where pi(θ) and qi(θ) are cubic trigonometric polynomials. The RCTFIF involves a scaling factor and four shape parameters in each subinterval. The convergence of the RCTFIF towards the original function is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the (Formula presented.)-RCTFIF so that the proposed RCTFIF preserves the positivity property of the given positive data set. 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 effects of the scaling factors and shape parameters on the RCTFIF and its first derivative are illustrated graphically.
Fractal Convolution Bessel Sequences on Rectangle
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.
Quintic hermite fractal interpolation in a strip: Preserving copositivity
The notion of fractal interpolation provides a general framework which includes traditional nonrecursive splines as special cases. In this paper, we describe a procedure for the construction of quintic Hermite FIFs as α-fractal function corresponding to the classical quintic Hermite interpolant. In contrast to traditional piecewise nonrecursive quintic Hermite interpolant, its fractal version has a second derivative which is differentiable in a finite or dense subset of the interpolation interval. This scheme offers an additional freedom over the classical quintic Hermite interpolants due to the presence of scaling factors. The elements of the iterated function system are identified so that the class of α-fractal function fα reflects the fundamental shape properties such as positivity, monotonicity, and convexity in addition to the regularity of f in the given interval. Using this general theory, an algorithm for positivity of quintic Hermite FIF is presented. Finally, the algorithm for a quintic Hermite fractal interpolants copositive with a given data set is prescribed.
A monotonic rational fractal interpolation surface and its analytical properties
A (Formula presented.)-continuous rational cubic fractal interpolation function was introduced and its monotonicity aspect was investigated in [Adv. Difference Eq. (30) 2014]. Using this univariate interpolant and a blending technique, in this article, we develop a monotonic rational fractal interpolation surface (FIS) for given monotonic surface data arranged on the rectangular grid. The analytical properties like convergence and stability of the rational cubic FIS are studied. Under some suitable hypotheses on the original function, the convergence of the rational cubic FIS is studied by calculating an upper bound for the uniform error of the surface interpolation. The stability results are studied when there is a small perturbation in the corresponding scaling factors. We also provide numerical examples to corroborate our theoretical results.