A Method of the Best Approximation by Fractal Function

Yong-Suk Kang; Myong-Gil Rim

doi:doi:10.11648/j.ijtam.20170301.12

| Peer-Reviewed

A Method of the Best Approximation by Fractal Function

Yong-Suk Kang, Myong-Gil Rim

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1)

Received: 7 September 2016 Accepted: 5 November 2016 Published: 9 December 2016

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1)
DOI	10.11648/j.ijtam.20170301.12
Page(s)	11-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Fractal Interpolation, Fractal Approximation, Iterated Function System, Fractal Function Space

References

[1]	M. F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988.
[2]	M. F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2(1) (1986) 303–329.
[3]	M. F. Barnsley, J. H. Elton, D. P. Hardin, Recurrent iterated function systems, Constr. Approx. 5(1) (1989) 3–31.
[4]	P. Bouboulis, L. Dalla, A general construction of fractal interpolation functions ongridsof Rⁿ, European J. Appl. Math. 18(4) (2007) 449–476.
[5]	P. Bouboulis, L. Dalla, Fractal interpolation surfaces derived from fractal interpolationfunctions, J. Math. Anal. Appl. 336(2) (2007) 919–936.
[6]	P. Bouboulis, L. Dalla, V. Drakopoulos Construction of recurrent bivariate fractal interpolation surfaces and computation of their boxcounting dimension, J. Approx. Theory 141 (2006) 99–117.
[7]	P. Bouboulis, M. Mavroforakis, Reproducing kernel Hilbert spaces and fractal interpolation, J. Comput. Appl. Math. 235 (2011) 3425–3434.
[8]	A. K. B. Chand, G. P. Kapoor, Generalized cubic spline fractal interpolation functions, SIAM J. Numer. Anal. 44(2) (2006) 655–676.
[9]	L. Dalla, Bivariated fractal interpolation functions on grids, Fractals 10(1) (2002) 53-58.
[10]	Z. G. Feng, Y. Z. Feng, Z. Y. Yuan, Fractal interpolation surfaces with function verticalscaling factors, Appl. Math. Lett. 25(11) (2012) 1896–1900.
[11]	R. Malysz, The Minkowski dimension of the bivariate fractal interpolation surfaces, Chaos Solitons Fractals 27(5) (2006) 1147–1156.
[12]	P. R. Massopust, Fractal Functions and their applications, Chaos Solitons Fractals 8(2) (1997) 171–190.
[13]	W. Metzler, C. H. Yun, Construction of fractal interpolation surfaces on rectangulargrids, Internat. J. Bifur. Chaos 20(12) (2010) 4079–4086.
[14]	M. A. Navascues, M. V. Sebastian, Generalization of Hermite functions by fractalinterpolation, J. Approx. Theory 131(1) (2004) 19–29.
[15]	S. Lonardi, P. Sommaruga, Fractal image approximation and orthogonal bases, SignalProcess. Image Commun. 14(5) (1999) 413–423.
[16]	Y. S. Kang, C. H. Yun, A construction of best fractal approximation, Electron. J. Math. Anal. Appl. 2(2) (2014) 144–151.
[17]	C. H. Yun, H. C. Choi, H. C. O, Construction of fractal surfaces by recurrent fractal interpolation curves, Chaos Solitons Fractals 66(2014) 136–143.
[18]	H. Zhang, R. Tao, S. Zhou, Y. Wang, Wavelet-based fractal function approxmation,J. Syst. Engrg. Electron. 10(4) (1999) 60–66.

Cite This Article

Plain Text BibTeX RIS

APA Style

Yong-Suk Kang, Myong-Gil Rim. (2016). A Method of the Best Approximation by Fractal Function. International Journal of Theoretical and Applied Mathematics, 3(1), 11-18. https://doi.org/10.11648/j.ijtam.20170301.12

Copy | Download

ACS Style

Yong-Suk Kang; Myong-Gil Rim. A Method of the Best Approximation by Fractal Function. Int. J. Theor. Appl. Math. 2016, 3(1), 11-18. doi: 10.11648/j.ijtam.20170301.12

Copy | Download

AMA Style

Yong-Suk Kang, Myong-Gil Rim. A Method of the Best Approximation by Fractal Function. Int J Theor Appl Math. 2016;3(1):11-18. doi: 10.11648/j.ijtam.20170301.12

Copy | Download

@article{10.11648/j.ijtam.20170301.12,
  author = {Yong-Suk Kang and Myong-Gil Rim},
  title = {A Method of the Best Approximation by Fractal Function},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {1},
  pages = {11-18},
  doi = {10.11648/j.ijtam.20170301.12},
  url = {https://doi.org/10.11648/j.ijtam.20170301.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170301.12},
  abstract = {We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Method of the Best Approximation by Fractal Function
AU  - Yong-Suk Kang
AU  - Myong-Gil Rim
Y1  - 2016/12/09
PY  - 2016
N1  - https://doi.org/10.11648/j.ijtam.20170301.12
DO  - 10.11648/j.ijtam.20170301.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 11
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170301.12
AB  - We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.
VL  - 3
IS  - 1
ER  -

Copy | Download

Author Information

Yong-Suk Kang

Faculty of Mathematics, Kim Il Sung University, Pyongyang, DPR Korea
Myong-Gil Rim

Faculty of Mathematics, Kim Il Sung University, Pyongyang, DPR Korea

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Yong-Suk Kang, Myong-Gil Rim. (2016). A Method of the Best Approximation by Fractal Function. International Journal of Theoretical and Applied Mathematics, 3(1), 11-18. https://doi.org/10.11648/j.ijtam.20170301.12

Copy | Download

ACS Style

Yong-Suk Kang; Myong-Gil Rim. A Method of the Best Approximation by Fractal Function. Int. J. Theor. Appl. Math. 2016, 3(1), 11-18. doi: 10.11648/j.ijtam.20170301.12

Copy | Download

AMA Style

Yong-Suk Kang, Myong-Gil Rim. A Method of the Best Approximation by Fractal Function. Int J Theor Appl Math. 2016;3(1):11-18. doi: 10.11648/j.ijtam.20170301.12

Copy | Download

@article{10.11648/j.ijtam.20170301.12,
  author = {Yong-Suk Kang and Myong-Gil Rim},
  title = {A Method of the Best Approximation by Fractal Function},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {1},
  pages = {11-18},
  doi = {10.11648/j.ijtam.20170301.12},
  url = {https://doi.org/10.11648/j.ijtam.20170301.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170301.12},
  abstract = {We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - A Method of the Best Approximation by Fractal Function
AU  - Yong-Suk Kang
AU  - Myong-Gil Rim
Y1  - 2016/12/09
PY  - 2016
N1  - https://doi.org/10.11648/j.ijtam.20170301.12
DO  - 10.11648/j.ijtam.20170301.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 11
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170301.12
AB  - We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.
VL  - 3
IS  - 1
ER  -

Copy | Download