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.

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.

Hidden variable bivariate fractal interpolation functions and errors on perturbations of function vertical scaling factors

2019 ◽  
Vol 12 (02) ◽  
pp. 1950021 ◽  
Author(s):  
Chol-Hui Yun ◽  
Mi-Kyong Ri
Keyword(s):  
Iterated Function System ◽  
Scaling Factor ◽  
Function System ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

In this paper, we present a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with function vertical scaling factors and estimate errors of HVBFIFs on perturbation of the function vertical scaling factor. We construct HVBFIFs on the basis of the iterated function system (IFS) with function vertical scaling factors. The perturbation of the function vertical scaling factors in the IFS causes a change in the HVBFIF. An upper estimation of the errors between the original HVBFIF and the perturbed HVBFIF is given.

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.

scholarly journals ANALYTIC PROPERTIES OF HIDDEN VARIABLE BIVARIABLE FRACTAL INTERPOLATION FUNCTIONS WITH FOUR FUNCTION CONTRACTIVITY FACTORS

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950018 ◽  
Author(s):  
CHOL-HUI YUN ◽  
MI-KYONG LI
Keyword(s):  
Experimental Data ◽  
Data Fitting ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Self Similarity ◽  
Scaling Factors ◽  
Small Perturbations ◽  
Hidden Variable ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

In this paper, we present some analytic properties of hidden variable bivariable fractal interpolation functions (HVBFIFs) with four function contractivity factors presented in [C. H. Yun and M. K. Li, Hidden variable bivariate fractal interpolation functions and errors on perturbations of function vertical scaling factors, Asian-Eur. J. Math. (2017), doi:10.1142/s1793557119500219]. Since four contractivity factors of these HVBFIFs are all functions, the construction of these HVBFIFs has more flexibility and diversity in fitting and approximation of complicated surfaces in nature and irregular experimental data with less self-similarity than one whose four contractivity factors are all constants or only one factor is function. The smoothness and stability of HVBFIFs are needed to ensure the applicability of the HVBFIFs in many practical problems such as the simulation of the objects of the nature, data fitting, etc. We first obtain the results related to their smoothness in nine different cases and then prove that the HVBFIFs are stable to the small perturbations of the interpolation points.

NONLINEAR RECURRENT HIDDEN VARIABLE FRACTAL INTERPOLATION CURVES WITH FUNCTION VERTICAL SCALING FACTORS

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050096
Author(s):  
JINMYONG KIM ◽  
HAKMYONG MUN
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Iterated Function Systems ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Hidden Variable ◽  
Iterated Function

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

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.

scholarly journals Multiresolution Analysis Based on Coalescence Hidden-Variable Fractal Interpolation Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
G. P. Kapoor ◽  
Srijanani Anurag Prasad
Keyword(s):  
Vector Space ◽  
Multiresolution Analysis ◽  
Riesz Bases ◽  
Fractal Interpolation ◽  
Hidden Variable ◽  
Free Variables ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions ◽  
Vector Subspaces

Multiresolution analysis arising from Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is developed. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in L2(R) than that provided by multiresolution analysis based only on Affine Fractal Interpolation Functions (AFIFs). Our approach consists of introduction of the vector space of CHFIFs, determination of its dimension and construction of Riesz bases of vector subspaces Vk, k∈Z, consisting of certain CHFIFs in L2(R)∩C0(R).

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