scholarly journals On Self‐Affine and Self‐Similar Graphs of Fractal Interpolation Functions Generated from Iterated Function Systems

2017 ◽  
Author(s):  
Sean Dillon ◽  
Vasileios Drakopoulos
Keyword(s):  
Iterated Function Systems ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Self Similar ◽  
Interpolation Functions

SURFACE FITTING AND ERROR ANALYSIS USING FRACTAL INTERPOLATION

2012 ◽  
Vol 22 (08) ◽  
pp. 1250194 ◽  
Author(s):  
HONG-YONG WANG ◽  
JIA-BING JI
Keyword(s):  
Error Analysis ◽  
Iterated Function Systems ◽  
Surface Fitting ◽  
Continuous Functions ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Continuous Surface ◽  
Fractal Interpolation Surface ◽  
Interpolation Functions

The fitting of a given continuous surface defined on a rectangular region in ℝ2 is studied by using a fractal interpolation surface, and the error analysis of fitting is made in this paper. The fractal interpolation functions used in surface fitting are generated by a special class of iterated function systems. Some properties of such fractal interpolation functions are discussed. Moreover, the error problems of fitting are investigated by using an operator defined on the space of continuous functions, and the upper estimates of errors are obtained in the sense of two kinds of metrics. Finally, a specific numerical example to illustrate the application of the procedure is also described.

ON SMOOTHNESS FOR A CLASS OF FRACTAL INTERPOLATION SURFACES

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 223-230 ◽  
Author(s):  
HONG-YONG WANG
Keyword(s):  
Wide Class ◽  
Iterated Function Systems ◽  
Fractal Interpolation ◽  
Refinement Equation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

In this paper, we consider a wide class of iterated function systems in R3, and show that their attractors are a class of fractal interpolation surfaces. Based on a refinement equation, we investigate the properties of smoothness of the fractal interpolation functions, and give the results of the smoothness in several cases.

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.

A general construction of fractal interpolation functions on grids of n

2007 ◽  
Vol 18 (4) ◽  
pp. 449-476 ◽  
Author(s):  
P. BOUBOULIS ◽  
L. DALLA
Keyword(s):  
Fractal Dimensions ◽  
Iterated Function Systems ◽  
Continuous Functions ◽  
Dirichlet Boundary Conditions ◽  
Data Sets ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Image Position ◽  
Interpolation Functions

We generalise the notion of fractal interpolation functions (FIFs) to allow data sets of the form where I=[0,1]n. We introduce recurrent iterated function systems whose attractors G are graphs of continuous functions f:I→, which interpolate the data. We show that the proposed constructions generalise the previously existed ones on . We also present some relations between FIFs and the Laplace partial differential equation with Dirichlet boundary conditions. Finally, the fractal dimensions of a class of FIFs are derived and some methods for the construction of functions of class Cp using recurrent iterated function systems are presented.

3D Fractal Interpolation Functions

2020 ◽  
Vol 12 (1) ◽  
pp. 120-123
Author(s):  
Rashad A. Al-Jawfi
Keyword(s):  
Data Analysis ◽  
Three Dimensional ◽  
Iterated Function Systems ◽  
Data Reconstruction ◽  
Fractal Interpolation ◽  
Scalar Data ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

For visualizing one, two- and/or three-dimensional data, reconstruction mechanism is generally considered. In the present study, I have examined the application of iterated function systems in interpolation. I have also presented new fractal interpolation derivations for three-dimensional scalar data. The interpolations can indicate uncertainty of the data, allow tenability from the data, represent the data statistically at different scales, and may also more accurately facilitate data analysis.

CONSTRUCTION OF NONLINEAR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR STABILITY

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950103
Author(s):  
JINMYONG KIM ◽  
HYONJIN KIM ◽  
HAKMYONG MUN
Keyword(s):  
Iterated Function Systems ◽  
Continuous Functions ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
The Stability ◽  
And Function ◽  
Stability Results ◽  
Interpolation Functions

This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.

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.

SENSITIVITY ANALYSIS FOR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR MOMENTS

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 161-170
Author(s):  
HONG-YONG WANG
Keyword(s):  
Sensitivity Analysis ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
The Difference ◽  
The Moment ◽  
Upper Estimate ◽  
Interpolation Functions

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

FRACTAL INTERPOLATION FUNCTIONS ON POST CRITICALLY FINITE SELF-SIMILAR SETS

Fractals ◽  
2010 ◽  
Vol 18 (01) ◽  
pp. 119-125 ◽  
Author(s):  
HUO-JUN RUAN
Keyword(s):  
Dirichlet Problem ◽  
Finite Energy ◽  
Sufficient Condition ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Self Similar ◽  
Interpolation Functions

In this paper, we introduce fractal interpolation functions (FIFs) and linear FIFs on a post critically finite (p.c.f. for short) self-similar set K. We present a sufficient condition such that linear FIFs have finite energy and prove that the solution of Dirichlet problem -Δμ u = f,u|∂K = 0 is a linear FIF on K if f is a linear FIF.

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