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.

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.

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.

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.

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].

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.

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

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.

A FRACTAL INTERPOLATION FUNCTION THROUGH SHEAR FACTOR WITH PERTURBATION ERROR ANALYSIS

2020 ◽  
Vol XVII (2) ◽  
pp. 15-22
Author(s):  
Tayba Arooj ◽  
Farsia Hussain ◽  
Malik Zawwar Hussain
Keyword(s):  
Iterated Function System ◽  
Vertical Shear ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
New Approach ◽  
Irregular Structure ◽  
Fractal Interpolation Function ◽  
Practical Usefulness ◽  
Iterated Function ◽  
Irregular Data

Fractal interpolation (FI) is widely used in nonlinear issues of natural and social sciences as it provides a constructive way to describe an irregular structure of data. A new approach of FI function (FIF) is proposed in this paper to represent uncertainty of irregular data. The proposed method is based on vertical shear factor instead of vertical scaling factor which is used in existing FI. Experiments were performed using existing datasets which confirmed the practical usefulness of the proposed method. A comparison with existing methods was also made to further verify the effectiveness of the proposed method. Furthermore, the criteria of the perturbed iterated function system (IFS) is proposed in order to satisfy the FI conditions and was explained with the help of an example. In addition, the error estimation between two IFS using the proposed refined equation is also presented.

scholarly journals Spline coalescence hidden variable fractal interpolation functions

2006 ◽  
Vol 2006 ◽  
pp. 1-17 ◽  
Author(s):  
A. K. B. Chand ◽  
G. P. Kapoor
Keyword(s):  
Iterated Function System ◽  
Hidden Variables ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
New Construction ◽  
Interpolation Functions

This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (1989), when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.

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