scholarly journals A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 767
Author(s):  
Alexandra Băicoianu ◽  
Cristina Maria Păcurar ◽  
Marius Păun
Keyword(s):  
Approximation Method ◽  
Finite System ◽  
Countable System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Finite Systems ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Finite Set ◽  
Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function(FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar .

INTEGRATION AND FOURIER TRANSFORM OF FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2005 ◽  
Vol 13 (01) ◽  
pp. 33-41 ◽  
Author(s):  
ZHIGANG FENG ◽  
LIXIN TIAN ◽  
JIANLI JIAO
Keyword(s):  
Fourier Transform ◽  
Continuous Function ◽  
Special Kind ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

Fractal interpolation function (FIF) is continuous on its interval of definition. As a special kind of continuous function, FIFs' integrations on various scales and Fourier transform are studied in this paper. All of them can be expressed by the parameters of the corresponding iterative function systems.

SOME RESULTS OF CONVERGENCE OF CUBIC SPLINE FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2003 ◽  
Vol 11 (01) ◽  
pp. 1-7 ◽  
Author(s):  
M. ANTONIA NAVASCUÉS ◽  
M. VICTORIA SEBASTIÁN
Keyword(s):  
Experimental Data ◽  
Cubic Spline ◽  
Spline Functions ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Original Function ◽  
New Methods ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).

AUTOCOVARIANCE AND INCREMENTS OF DEVIATION OF FRACTAL INTERPOLATION FUNCTIONS FOR RANDOM DATASETS

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850075
Author(s):  
DAH-CHIN LUOR
Keyword(s):  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Random Dataset ◽  
Interpolation Functions

In this paper we consider the expectation, the autocovariance, and increments of the deviation of a fractal interpolation function [Formula: see text] corresponding to a random dataset [Formula: see text]. We show that the covariance of [Formula: see text] and [Formula: see text] is a fractal interpolation function on [Formula: see text] for each fixed [Formula: see text], where [Formula: see text]. We also prove that, for a fixed [Formula: see text], the covariance of [Formula: see text] and [Formula: see text] is a fractal interpolation function on [Formula: see text]. A special type of increments of the deviation of [Formula: see text] is also investigated.

scholarly journals CUBIC SPLINE SUPER FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2014 ◽  
Vol 22 (01n02) ◽  
pp. 1450005 ◽  
Author(s):  
G. P. KAPOOR ◽  
SRIJANANI ANURAG PRASAD
Keyword(s):  
Generating Function ◽  
Cubic Spline ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Approximation Properties ◽  
Fractal Interpolation Function ◽  
Convergence Results ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF[Formula: see text] and their derivatives [Formula: see text] converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.

scholarly journals Graph-directed random fractal interpolation function

2021 ◽  
Vol 66 (2) ◽  
pp. 247-255
Author(s):  
Ildiko Somogyi ◽  
Anna Soos
Keyword(s):  
Real Data ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractal Function ◽  
Stochastic Version ◽  
Fractal Interpolation Function ◽  
Random Fractal ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and  Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."

scholarly journals The Smoothness of Fractal Interpolation Functions onℝand onp-Series Local Fields

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jing Li ◽  
Weiyi Su
Keyword(s):  
Local Field ◽  
Fractal Dimensions ◽  
Local Fields ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractal Function ◽  
Type Function ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

A fractal interpolation function on ap-series local fieldKpis defined, and itsp-type smoothness is shown by virtue of the equivalent relationship between the Hölder type spaceCσKpand the Lipschitz class Lipσ,Kp. The orders of thep-type derivatives and the fractal dimensions of the graphs of Weierstrass type function on local fields are given as an example. Theα-fractal function onℝis introduced and the conclusion of its smoothness is improved in a more general case; some examples are shown to support the conclusion. Finally, a comparison between the fractal interpolation functions defined onℝandKpis given.

FRACTIONAL CALCULUS OF COALESCENCE HIDDEN-VARIABLE FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750019 ◽  
Author(s):  
SRIJANANI ANURAG PRASAD
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Free Parameters ◽  
Interpolation Functions

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order [Formula: see text] of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF.

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 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.

Sign in / Sign up

Export Citation Format

Share Document