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.

HIDDEN VARIABLE RECURRENT FRACTAL INTERPOLATION FUNCTIONS WITH FUNCTION CONTRACTIVITY FACTORS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950113
Author(s):  
CHOL-HUI YUN
Keyword(s):  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Better Than ◽  
Interpolation Functions

In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIFs) with four function contractivity factors. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed using a recurrent iterated function system (RIFS). In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation functions. RIFSs produce fractal sets with local self-similarity structure. Therefore, the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). The HVFIF is neither self-similar nor self-affine one. Hence, the HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The contractivity factors of IFS are very important one that determines characteristics of FIFs. The IFS and RIFS with function contractivity factors can describe the fractal objects in nature better than one with constant contractivity factors. To ensure higher flexibility and diversity of the construction of the FIFs, we present constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

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.

scholarly journals Uniform Continuity of Fractal Interpolation Function

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Xuezai Pan ◽  
Minggang Wang ◽  
Xudong Shang
Keyword(s):  
Iterated Function System ◽  
Uniform Continuity ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Covering Theorem ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Definition Of ◽  
Uniformly Continuous ◽  
Research Analysis

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.

scholarly journals Fractal Interpolation Using Harmonic Functions on the Koch Curve

2021 ◽  
Vol 5 (2) ◽  
pp. 28
Author(s):  
Song-Il Ri ◽  
Vasileios Drakopoulos ◽  
Song-Min Nam
Keyword(s):  
Harmonic Functions ◽  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Koch Curve ◽  
Fractal Sets ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

THE PROPERTIES OF FRACTIONAL ORDER CALCULUS OF FRACTAL INTERPOLATION FUNCTION OF BROKEN LINE SEGMENTS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850031 ◽  
Author(s):  
XUEZAI PAN ◽  
RONGFEI XU ◽  
XUDONG SHANG ◽  
MINGGANG WANG
Keyword(s):  
Fractional Order ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractional Order Calculus ◽  
Line Segments ◽  
First Order ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractional Order Integral

In order to research the properties of the fractional order calculus of broken line segments’ fractal interpolation function (FIF) generated by the linear iterated function system (IFS), the concepts of the Riemann–Liouville fractional order calculus and the method of the IFS are used to prove the properties of the fractional calculus of the broken line segments’ FIF generated by the linear IFS. There are two conclusions as follows. First, the fractional order integral of the broken line segments’ FIF formed by the linear IFS is continuous and first-order differentiable on the closed interval [Formula: see text]. Second, the broken line segments’ FIF formed by the linear IFS exists with fractional order differential, but the differential function is not continuous.

scholarly journals The fractal interpolation for countable systems of data

2003 ◽  
pp. 11-19 ◽  
Author(s):  
Nicolae-Adrian Secelean
Keyword(s):  
Iterated Function System ◽  
Real Interval ◽  
Countable System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Finite Case ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Countable Iterated Function System ◽  
Arcwise Connected

In this paper we will extend the fractal interpolation from the finite case to the case of countable sets of data. The main result is that, given an countable system of data in [a, b] ? Y, where [a, b] is a real interval and Y a compact and arcwise connected metric space, there exists a countable iterated function system whose attractor is the graph of a fractal interpolation function.

scholarly journals Fractional Calculus of Fractal Interpolation Function on[0,b](b>0)

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
XueZai Pan
Keyword(s):  
Continuous Function ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractional Order Calculus ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractional Order Integral

The paper researches the continuity of fractal interpolation function’s fractional order integral on[0,+∞)and judges whether fractional order integral of fractal interpolation function is still a fractal interpolation function on[0,b](b>0)or not. Relevant theorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researched content. The conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on[0,+∞)and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval[0,b].

A REVISIT TO α-FRACTAL FUNCTION AND BOX DIMENSION OF ITS GRAPH

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950090
Author(s):  
S. VERMA ◽  
P. VISWANATHAN
Keyword(s):  
Fractal Geometry ◽  
Iterated Function System ◽  
Wide Spectrum ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Constructive Approximation

One of the tools offered by fractal geometry is fractal interpolation, which forms a basis for the constructive approximation theory for nondifferentiable functions. The notion of fractal interpolation function can be used to obtain a wide spectrum of self-referential functions associated to a prescribed continuous function on a compact interval in [Formula: see text]. These fractal maps, the so-called [Formula: see text]-fractal functions, are defined by means of suitable iterated function system which involves some parameters. Building on the literature related to the notion of [Formula: see text]-fractal functions, the current study targets to record the continuous dependence of the [Formula: see text]-fractal function on parameters involved in its definition. Furthermore, the paper attempts to study the box dimension of the graph of the [Formula: see text]-fractal function.

Hidden Variable Bivariate Fractal Interpolation Surfaces

Fractals ◽  
2003 ◽  
Vol 11 (03) ◽  
pp. 277-288 ◽  
Author(s):  
A. K. B. Chand ◽  
G. P. Kapoor
Keyword(s):  
Iterated Function System ◽  
Degree Of Freedom ◽  
Function System ◽  
Fractal Interpolation ◽  
Fractal Surface ◽  
Hidden Variable ◽  
Iterated Function ◽  
Self Similar ◽  
Vector Valued

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ4 and its projection in ℝ3 is taken. The extra degree of freedom coming from ℝ4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

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