Hölder Exponents and Box Dimension for Self-Affine Fractal Functions

1989 ◽  
pp. 33-48
Author(s):  
Tim Bedford
Keyword(s):  
Box Dimension ◽  
Fractal Functions ◽  
Hölder Exponents

FRACTAL DIMENSIONS OF LINEAR COMBINATION OF CONTINUOUS FUNCTIONS WITH THE SAME BOX DIMENSION

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050139
Author(s):  
XUEFEI Wang ◽  
CHUNXIA ZHAO
Keyword(s):  
Linear Combination ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
Fractal Function ◽  
Space Of Continuous Functions ◽  
Fractal Functions

In this paper, we mainly discuss continuous functions with certain fractal dimensions on [Formula: see text]. We find space of continuous functions with certain Box dimension is not closed. Furthermore, Box dimension of linear combination of two continuous functions with the same Box dimension maybe does not exist. Definitions of fractal functions and local fractal functions have been given. Linear combination of a fractal function and a local fractal function with the same Box dimension must still be the original Box dimension with nontrivial coefficients.

Novel approach to calculation of box dimension of fractal functions

2014 ◽  
Vol 8 ◽  
pp. 7175-7181
Author(s):  
Konstantin Igudesman ◽  
Roman Lavrenov ◽  
Victor Klassen
Keyword(s):  
Box Dimension ◽  
Novel Approach ◽  
Fractal Functions

scholarly journals Box dimension for graphs of fractal functions

1997 ◽  
Vol 40 (2) ◽  
pp. 331-344
Author(s):  
Gavin Brown ◽  
Qinghe Yin
Keyword(s):  
Iterated Function Systems ◽  
Box Dimension ◽  
Affine Maps ◽  
Iterated Function ◽  
Fractal Functions

We calculate the box-dimension for a class of nowhere differentiable curves defined by Markov attractors of certain iterated function systems of affine maps.

BOX DIMENSIONS OF α-FRACTAL FUNCTIONS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650037 ◽  
Author(s):  
MD. NASIM AKHTAR ◽  
M. GURU PREM PRASAD ◽  
M. A. NAVASCUÉS
Keyword(s):  
Continuous Function ◽  
Differentiable Function ◽  
Iterated Function System ◽  
Box Dimension ◽  
Data Sets ◽  
Fractal Function ◽  
Iterated Function ◽  
Nowhere Differentiable Function ◽  
Fractal Functions ◽  
Box Dimensions

The box dimension of the graph of non-affine, continuous, nowhere differentiable function [Formula: see text] which is a fractal analogue of a continuous function [Formula: see text] corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of [Formula: see text]-fractal function [Formula: see text] for equally spaced as well as arbitrary data sets are found.

On a class of fractal functions with graph Box dimension 2

2004 ◽  
Vol 22 (1) ◽  
pp. 135-139 ◽  
Author(s):  
T.F. Xie ◽  
S.P. Zhou
Keyword(s):  
Box Dimension ◽  
Fractal Functions ◽  
Dimension 2

PROGRESS ON ESTIMATION OF FRACTAL DIMENSIONS OF FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950084 ◽  
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
One Dimensional ◽  
Zero Measure ◽  
Liouville Fractional Derivative ◽  
Fractal Functions ◽  
Dimension One

In this paper, fractal dimensions of fractional calculus of continuous functions defined on [Formula: see text] have been explored. Continuous functions with Box dimension one have been divided into five categories. They are continuous functions with bounded variation, continuous functions with at most finite unbounded variation points, one-dimensional continuous functions with infinite but countable unbounded variation points, one-dimensional continuous functions with uncountable but zero measure unbounded variation points and one-dimensional continuous functions with uncountable and non-zero measure unbounded variation points. Box dimension of Riemann–Liouville fractional integral of any one-dimensional continuous functions has been proved to be with Box dimension one. Continuous functions on [Formula: see text] are divided as local fractal functions and fractal functions. According to local structure and fractal dimensions, fractal functions are composed of regular fractal functions, irregular fractal functions and singular fractal functions. Based on previous work, upper Box dimension of any continuous functions has been proved to be no less than upper Box dimension of their Riemann–Liouville fractional integral. Fractal dimensions of Riemann–Liouville fractional derivative of certain continuous functions have been investigated elementary.

A REVISIT TO α-FRACTAL FUNCTION AND BOX DIMENSION OF ITS GRAPH

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950090
Author(s):  
S. VERMA ◽  
P. VISWANATHAN
Keyword(s):  
Fractal Geometry ◽  
Iterated Function System ◽  
Wide Spectrum ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Function ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractal Functions ◽  
Constructive Approximation

One of the tools offered by fractal geometry is fractal interpolation, which forms a basis for the constructive approximation theory for nondifferentiable functions. The notion of fractal interpolation function can be used to obtain a wide spectrum of self-referential functions associated to a prescribed continuous function on a compact interval in [Formula: see text]. These fractal maps, the so-called [Formula: see text]-fractal functions, are defined by means of suitable iterated function system which involves some parameters. Building on the literature related to the notion of [Formula: see text]-fractal functions, the current study targets to record the continuous dependence of the [Formula: see text]-fractal function on parameters involved in its definition. Furthermore, the paper attempts to study the box dimension of the graph of the [Formula: see text]-fractal function.

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