scholarly journals Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
John Kieffer
Keyword(s):  
Initial Conditions ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Continuous Functions ◽  
Bounded Sequence ◽  
Divide And Conquer ◽  
Real Numbers ◽  
Fixed Integer ◽  
Iterated Function ◽  
International Audience

International audience Let $k≥2$ be a fixed integer. Given a bounded sequence of real numbers $(a_n:n≥k)$, then for any sequence $(f_n:n≥1)$ of real numbers satisfying the divide-and-conquer recurrence $f_n = (k-mod(n,k))f_⌊n/k⌋+mod(n,k)f_⌈n/k⌉ + a_n, n ≥k$, there is a unique continuous periodic function $f^*:\mathbb{R}→\mathbb{R}$ with period 1 such that $f_n = nf^*(\log _kn)+o(n)$. If $(a_n)$ is periodic with period $k, a_k=0$, and the initial conditions $(f_i:1 ≤i ≤k-1)$ are all zero, we obtain a specific iterated function system $S$, consisting of $k$ continuous functions from $[0,1]×\mathbb{R}$ into itself, such that the attractor of $S$ is $\{(x,f^*(x)): 0 ≤x ≤1\}$. Using the system $S$, an accurate plot of $f^*$ can be rapidly obtained.

scholarly journals Dimension spectrum of infinite self-affine iterated function systems

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Natalia Jurga
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Real Numbers ◽  
Dimension Spectrum ◽  
Iterated Function ◽  
Isolated Points ◽  
First Time

AbstractGiven an infinite iterated function system (IFS) $${\mathcal {F}}$$ F , we define its dimension spectrum $$D({\mathcal {F}})$$ D ( F ) to be the set of real numbers which can be realised as the dimension of some subsystem of $${\mathcal {F}}$$ F . In the case where $${\mathcal {F}}$$ F is a conformal IFS, the properties of the dimension spectrum have been studied by several authors. In this paper we investigate for the first time the properties of the dimension spectrum when $${\mathcal {F}}$$ F is a non-conformal IFS. In particular, unlike dimension spectra of conformal IFS which are always compact and perfect (by a result of Chousionis, Leykekhman and Urbański, Selecta 2019), we construct examples to show that $$D({\mathcal {F}})$$ D ( F ) need not be compact and may contain isolated points.

scholarly journals Iterated function systems consisting of continuous functions satisfying Banach’s orbital condition

2018 ◽  
Vol 56 (2) ◽  
pp. 71-80
Author(s):  
Radu Miculescu ◽  
Alexandru Mihail ◽  
Irina Savu
Keyword(s):  
Iterated Function System ◽  
Iterated Function Systems ◽  
Continuous Functions ◽  
Function System ◽  
Fractal Operator ◽  
Iterated Function

AbstractWe introduce the concept of iterated function system consisting of continuous functions satisfying Banach’s orbital condition and prove that the fractal operator associated to such a system is weakly Picard. Some examples are provided.

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.

scholarly journals Fractal convolution: A new operation between functions

2019 ◽  
Vol 22 (3) ◽  
pp. 619-643
Author(s):  
María A. Navascués ◽  
Peter R. Massopust
Keyword(s):  
Hilbert Spaces ◽  
Binary Operation ◽  
Iterated Function System ◽  
Continuous Functions ◽  
Linear Operators ◽  
Function System ◽  
Fractal Function ◽  
Iterated Function

Abstract In this paper, we define an internal binary operation between functions called fractal convolution that when applied to a pair of mappings generates a fractal function. This is done by means of a suitably defined iterated function system. We study in detail this operation in 𝓛p spaces and in sets of continuous functions in a way that is different from the previous work of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals with the construction of convolved fractals bases and frames in Banach and Hilbert spaces of functions.

scholarly journals MULTIPLE REPRESENTATIONS OF REAL NUMBERS ON SELF-SIMILAR SETS WITH OVERLAPS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950051 ◽  
Author(s):  
KAN JIANG ◽  
XIAOMIN REN ◽  
JIALI ZHU ◽  
LI TIAN
Keyword(s):  
Convex Hull ◽  
Iterated Function System ◽  
Multiple Representations ◽  
Function System ◽  
Real Numbers ◽  
Iterated Function ◽  
Self Similar

Let [Formula: see text] be the attractor of the following iterated function system (IFS) [Formula: see text] where [Formula: see text] and [Formula: see text] is the convex hull of [Formula: see text]. The main results of this paper are as follows: [Formula: see text] if and only if [Formula: see text] where [Formula: see text]. If [Formula: see text], then [Formula: see text]As a consequence, we prove that the following conditions are equivalent:(1) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text].(2) For any [Formula: see text], there are some [Formula: see text] such that [Formula: see text](3) [Formula: see text].

MULTI-DIMENSIONAL PIECE-WISE SELF-AFFINE FRACTAL INTERPOLATION MODEL IN TENSOR FORM

2006 ◽  
Vol 09 (03) ◽  
pp. 287-293 ◽  
Author(s):  
TONG ZHANG ◽  
JIANLIN LIU ◽  
ZHUO ZHUANG
Keyword(s):  
Partial Derivative ◽  
Iterated Function System ◽  
Model Parameters ◽  
Fractal Interpolation ◽  
Interpolation Model ◽  
Tensor Form ◽  
Real Numbers ◽  
Inverse Algorithm ◽  
Iterated Function ◽  
Discrete Sequences

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a "mapping partial derivative" and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in Rn. Therefore, the result is very general. Moreover, the multi-dimensional piece-wise self-affine fractal interpolation model in tensor form is more terse than in the usual matrix form.

GENERALIZATIONS OF THE KOCH CURVE

Fractals ◽  
2008 ◽  
Vol 16 (03) ◽  
pp. 267-274 ◽  
Author(s):  
R. B. DARST ◽  
J. A. PALAGALLO ◽  
T. E. PRICE
Keyword(s):  
Iterative Method ◽  
Iterated Function System ◽  
Continuous Functions ◽  
Cantor Sets ◽  
Koch Curve ◽  
Double Points ◽  
Iterated Function ◽  
Fixed Set ◽  
Equilateral Triangles ◽  
Two Parameter

We present an iterative method to define a two-parameter family of continuous functions fa,θ: I → ℂ such that f1/3,π/3 is the Koch curve. We consider the two-cases θ = π/3 and θ = π/4 of these generalized Koch curves fa,θ(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where θ = π/3 double points are vertices of equilateral triangles. When θ = π/4 the double points form Cantor sets in the plane. We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.

scholarly journals On fractal properties of Weierstrass-type functions

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Claire David
Keyword(s):  
Real Number ◽  
Iterated Function System ◽  
Function System ◽  
Box Dimension ◽  
Intrinsic Properties ◽  
Weierstrass Function ◽  
Real Numbers ◽  
New Class ◽  
Fractal Properties ◽  
Iterated Function

In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {\mathcal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left(2\, \pi\,N_b^n\,x \right)$, where $\lambda$ and $N_b$ are two real numbers such that~\mbox{$0 <\lambda<1$},~\mbox{$ N_b\,\in\,\N$} and $ \lambda\,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.

scholarly journals Generalized Iterated Function Systems Containing Functions of Integral Type

2018 ◽  
Vol 7 (3.31) ◽  
pp. 126
Author(s):  
Minirani S ◽  
. .
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Function System ◽  
Finite Collection ◽  
Integral Type ◽  
Iterated Function

A finite collection of mappings which are contractions on a complete metric space constitutes an iterated function system. In this paper we study the generalized iterated function system which contain generalized contractions of integral type from the product space . We prove the existence and uniqueness of the fixed point of such an iterated function system which is also known as its attractor. 

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