scholarly journals FRACTAL APPROXIMATION OF JACKSON TYPE FOR PERIODIC PHENOMENA

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850079 ◽  
Author(s):  
M. A. NAVASCUÉS ◽  
SANGITA JHA ◽  
A. K. B. CHAND ◽  
M. V. SEBASTIÁN
Keyword(s):  
Functional Model ◽  
Classical Case ◽  
Sampling Frequency ◽  
Trigonometric Polynomials ◽  
Approximation Formula ◽  
Fractal Interpolation ◽  
Trigonometric Functions ◽  
Fractal Functions ◽  
Fractal Approximation ◽  
Range Of Values

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators and their classical counterparts are proposed.

Some Results on Recurrent Fractal Interpolation Function

2020 ◽  
Vol 12 (8) ◽  
pp. 1038-1043
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Data Set ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Fractal Approximation

Barnsley (Barnsley, M.F., 1986. Fractal functions and interpolation. Constr. Approx., 2, pp.303–329) introduced fractal interpolation function (FIF) whose graph is the attractor of an iterated function system (IFS) for describing the data that have an irregular or self-similar structure. Barnsley et al. (Barnsley, M.F., et al., 1989. Recurrent iterated function systems in fractal approximation. Constr. Approx., 5, pp.3–31) generalized FIF in the form of recurrent fractal interpolation function (RFIF) whose graph is the attractor of a recurrent iterated function system (RIFS) to fit data set which is piece-wise self-affine. The primary aim of the present research is investigating the RFIF approach and using it for fitting the piece-wise self-affine data set in ℜ2.

FRACTAL TRIGONOMETRIC POLYNOMIALS FOR RESTRICTED RANGE APPROXIMATION

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650022 ◽  
Author(s):  
A. K. B. CHAND ◽  
M. A. NAVASCUÉS ◽  
P. VISWANATHAN ◽  
S. K. KATIYAR
Keyword(s):  
Continuous Functions ◽  
Trigonometric Polynomials ◽  
Fractal Interpolation ◽  
Restricted Range ◽  
Interpolation Function ◽  
Trigonometric Functions ◽  
Fractal Interpolation Function

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.

PARTIALLY BLENDED CONSTRAINED RATIONAL CUBIC TRIGONOMETRIC FRACTAL INTERPOLATION SURFACES

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650027 ◽  
Author(s):  
A. K. B. CHAND ◽  
K. R. TYADA
Keyword(s):  
Trigonometric Polynomials ◽  
Fractal Interpolation ◽  
Shape Parameters ◽  
New Family ◽  
Surface Data ◽  
Fractal Interpolation Functions ◽  
Blending Functions ◽  
Special Case ◽  
Theoretical Results ◽  
Interpolation Functions

Fractal interpolation is an advance technique for visualization of scientific shaped data. In this paper, we present a new family of partially blended rational cubic trigonometric fractal interpolation surfaces (RCTFISs) with a combination of blending functions and univariate rational trigonometric fractal interpolation functions (FIFs) along the grid lines of the interpolation domain. The developed FIFs use rational trigonometric functions [Formula: see text], where [Formula: see text] and [Formula: see text] are cubic trigonometric polynomials with four shape parameters. The convergence analysis of partially blended RCTFIS with the original surface data generating function is discussed. We derive sufficient data-dependent conditions on the scaling factors and shape parameters such that the fractal grid line functions lie above the grid lines of a plane [Formula: see text], and consequently the proposed partially blended RCTFIS lies above the plane [Formula: see text]. Positivity preserving partially blended RCTFIS is a special case of the constrained partially blended RCTFIS. Numerical examples are provided to support the proposed theoretical results.

scholarly journals Multivariate Fractal Functions in Some Complete Function Spaces and Fractional Integral of Continuous Fractal Functions

2021 ◽  
Vol 5 (4) ◽  
pp. 185
Author(s):  
Kshitij Kumar Pandey ◽  
Puthan Veedu Viswanathan
Keyword(s):  
Fractional Integral ◽  
Function Spaces ◽  
Fractal Interpolation ◽  
Simple Consequence ◽  
Fractal Function ◽  
Lebesgue Spaces ◽  
Fractal Interpolation Function ◽  
Fractal Operator ◽  
Fractal Functions ◽  
Similar Kind

There has been a considerable evolution of the theory of fractal interpolation function (FIF) over the last three decades. Recently, we introduced a multivariate analogue of a special class of FIFs, which is referred to as α-fractal functions, from the viewpoint of approximation theory. In the current note, we continue our study on multivariate α-fractal functions, but in the context of a few complete function spaces. For a class of fractal functions defined on a hyperrectangle Ω in the Euclidean space Rn, we derive conditions on the defining parameters so that the fractal functions are elements of some standard function spaces such as the Lebesgue spaces Lp(Ω), Sobolev spaces Wm,p(Ω), and Hölder spaces Cm,σ(Ω), which are Banach spaces. As a simple consequence, for some special choices of the parameters, we provide bounds for the Hausdorff dimension of the graph of the corresponding multivariate α-fractal function. We shall also hint at an associated notion of fractal operator that maps each multivariate function in one of these function spaces to its fractal counterpart. The latter part of this note establishes that the Riemann–Liouville fractional integral of a continuous multivariate α-fractal function is a fractal function of similar kind.

SHAPE PRESERVING RATIONAL QUARTIC FRACTAL FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950141 ◽  
Author(s):  
S. K. KATIYAR ◽  
A. K. B. CHAND
Keyword(s):  
Iterated Function System ◽  
Linear Equations ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Shape Preserving ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Fractal Functions ◽  
Interpolation Functions

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct [Formula: see text]-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the [Formula: see text]-fractal rational quartic spline when the original function is in [Formula: see text]. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the [Formula: see text]-fractal rational quartic spline to [Formula: see text]. The elements of the iterated function system are identified befittingly so that the class of [Formula: see text]-fractal function [Formula: see text] incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ [Formula: see text]. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

scholarly journals Non-Stationary Fractal Interpolation

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 666 ◽  
Author(s):  
Peter Massopust
Keyword(s):  
Fixed Points ◽  
Iterated Function System ◽  
Function System ◽  
Global Behavior ◽  
Fractal Interpolation ◽  
The Novel ◽  
Countable Sequence ◽  
Iterated Function ◽  
Fractal Functions ◽  
Novel Concept

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k ∈ N where each F k maps H ( X ) → H ( X ) and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.

A FRACTAL APPROXIMATION TO PERIODICITY

Fractals ◽  
2006 ◽  
Vol 14 (04) ◽  
pp. 315-325 ◽  
Author(s):  
M. A. NAVASCUÉS
Keyword(s):  
Iterated Function Systems ◽  
Trigonometric Polynomials ◽  
Small Oscillations ◽  
One Dimensional ◽  
Iterated Function ◽  
Fractal Approximation ◽  
The One ◽  
Circular Functions ◽  
Nature And Society ◽  
Analytical Tools

Periodicity is recurrent in nature and society. The problem of the periodicity is faced here with the support of fractal methodology. Some analytical tools for the understanding of the one-dimensional projections of periodic phenomena are proposed. The reconstruction of an unknown periodic sampled variable is approached, assuming a deterministic self-affine nature in its small oscillations. Fractal trigonometric polynomials are defined by means of suitable iterated function systems. These objects are fractal perturbations of the classical circular functions. The coefficients of the system enable the control and modification of the properties of the originals. Additionally, Fourier parameters and approximants for sampled signals are computed and the density of the fractal trigonometric polynomials in the most common spaces of periodic functions is proved.

scholarly journals New Approach to Fractal Approximation of Vector-Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Konstantin Igudesman ◽  
Marsel Davletbaev ◽  
Gleb Shabernev
Keyword(s):  
Fractal Interpolation ◽  
New Approach ◽  
Smooth Curves ◽  
Continuous Vector ◽  
Multidimensional Generalization ◽  
Fractal Interpolation Functions ◽  
Fractal Approximation ◽  
Vector Sequences ◽  
Interpolation Functions

This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions which are multidimensional generalization of fractal interpolation functions. Best values of fractal interpolation vector-functions parameters are found. We give schemes of approximation of some sets of data and consider examples of approximation of smooth curves with different conditions.

scholarly journals Local Fractal Interpolation on Unbounded Domains

2018 ◽  
Vol 61 (1) ◽  
pp. 151-167 ◽  
Author(s):  
Peter R. Massopust
Keyword(s):  
Unbounded Domains ◽  
Tensor Products ◽  
Topological Spaces ◽  
Fractal Interpolation ◽  
Lebesgue Spaces ◽  
Fractal Functions

AbstractWe define fractal interpolation on unbounded domains for a certain class of topological spaces and construct local fractal functions. In addition, we derive some properties of these local fractal functions, consider their tensor products, and give conditions for local fractal functions on unbounded domains to be elements of Bochner–Lebesgue spaces.

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