A Concretization of an Approximation Method for NonAffine Fractal Interpolation Functions
Abstract
:1. Introduction
2. Mathematical Preliminaries
 1.
 A map $f:X\to X$ is called Lipschitz if there exists a real nonnegative C such that$$d\left(f\right(x),f(y\left)\right)\le Cd(x,y),$$for every $x,y\in X$. The smallest C in the above definition is called Lipschitz constant and it is defined as$$lip\left(f\right)=\underset{x\ne y}{sup}{\displaystyle \frac{d\left(f\right(x),f(y\left)\right)}{d(x,y)}}$$
 2.
 A map $f:X\to X$ is called Banach contraction if there exists $C\in (0,1)$ such that$$d\left(f\right(x),f(y\left)\right)\le Cd(x,y),$$for every $x,y\in X$.
 3.
 A map $f:X\to X$ is called φcontraction if there exists a function $\phi :[0,\infty )\to [0,\infty )$ such that$$d\left(f\right(x),f(y\left)\right)\le \phi \left(d\right(x,y\left)\right),$$for every $x,y\in X$.
 4.
 A map $f:X\to X$ is called Matkowski contraction if it is a φcontraction where $\phi :[0,\infty )\to [0,\infty )$ is nondecreasing and $\underset{n\to \infty}{lim}{\phi}^{\left[n\right]}\left(t\right)=0$ for all $t>0$.
 5.
 A map $f:X\to X$ is called Rakotch contraction if it is a φcontraction where $\phi :[0,\infty )\to [0,\infty )$ is such that the function $t\to \frac{\phi \left(t\right)}{t}$ is nonincreasing for every $t>0$ and $\frac{\phi \left(t\right)}{t}<1$ for every $t\in (0,\infty )$.
 1.
 Every Banach contraction is Lipschitz where the Lipschitz constant is smaller than 1.
 2.
 Every Banach contraction is a φcontraction, for$$\phi \left(t\right)=C\xb7t,$$for every $t>0$.
 3.
 Every Rakotch contraction is a Matkowski contraction.
2.1. Iterated Function Systems
2.2. Countable FIFs
 (i)
 there exists ${C}_{n}\in [0,1)$ such that$${l}_{n}\left(x\right){l}_{n}\left({x}^{\prime}\right)\le {C}_{n}x{x}^{\prime}$$
 (ii)
 $${l}_{n}\left({x}_{0}\right)={x}_{n1}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{l}_{n}\left(m\right)={x}_{n};$$
 (iii)
 $$\underset{n\ge 1}{sup}\phantom{\rule{0.166667em}{0ex}}{C}_{n}<1.$$
 (j)
 $${F}_{n}({x}_{0},{y}_{0})={y}_{n1}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{F}_{n}(m,M)={y}_{n};$$
 (jj)
 $\underset{n\to \infty}{lim}diam\left(Im\phantom{\rule{0.166667em}{0ex}}{F}_{n}\right)=0$.
 1.
 If the functions ${F}_{n}$ are Lipschitz with respect to the first variable and Rakotch contractions with respect to the second variable, then the functions ${f}_{n}$ are Rakotch contractions with respect to ${d}_{\theta}$, where$${d}_{\theta}((x,y),({x}^{\prime},{y}^{\prime})):=x{x}^{\prime}+\theta d(y,{y}^{\prime})$$for all $(x,y),\phantom{\rule{0.166667em}{0ex}}({x}^{\prime},{y}^{\prime})\in [{x}_{0},m]\times Y$, where $\theta =\frac{1\underset{n\ge 1}{sup}{C}_{n}}{2(C+1)}\in (0,1)$.
 2.
 Given the same aforementioned framework, there exists an interpolation function ${f}_{*}$ corresponding to the system of data (1) such that its graph is the attractor of the CIFS $\mathcal{S}=(([{x}_{0},m]\times Y,{d}_{\theta}),{\left({f}_{n}\right)}_{n\ge 1})$.
3. Computational Background
3.1. Applied Technologies. Motivation (Pros)
3.2. Technical Notes on Performance
3.3. Limitations (Constraints)
4. Main Results
4.1. Countable Fractal NonAffine Interpolation Schemes
Algorithm 1: The Probabilistic scheme. 

Algorithm 2: The Deterministic Scheme 

4.2. Countable Fractal Affine Interpolation Schemes
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
 Barnsley, M.F. Fractal functions and interpolation. Constr. Approx. 1986, 2, 303–329. [Google Scholar] [CrossRef]
 Barnsley, M. Fractals Everywhere; Academic Press: New York, NY, USA, 1988. [Google Scholar]
 Barnsley, M.F.; Elton, J.; Hardin, D.; Massopust, P. Hidden variable fractal interpolation functions. SIAM J. Math. Anal. 1989, 20, 1218–1242. [Google Scholar] [CrossRef]
 Mazel, D.S.; Hayes, M.H. Hiddenvariable fractal interpolation of discrete sequences. In ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing; IEEE Computer Society: Toronto, ON, Canada, 1991. [Google Scholar]
 Chand, A.K.B.; Kapoor, G.P. Hidden variable bivariate fractal interpolation surfaces. Fractals 2003, 11, 277–288. [Google Scholar] [CrossRef]
 Bouboulis, P.; Dalla, L. Hidden variable vector valued fractal interpolation functions. Fractals 2005, 13, 227–232. [Google Scholar] [CrossRef]
 Fernau, H. Infinite iterated function systems. Math. Nachrichten 1994, 170, 79–91. [Google Scholar] [CrossRef] [Green Version]
 Secelean, N. Countable Iterated Fuction Systems. Far East J. Dym. Syst. 2001, 3, 149–167. [Google Scholar]
 Secelean, N. Countable Iterated Function Systems; LAP Lambert Academic Publishing: Saarbrueken, Germany, 2013. [Google Scholar]
 Secelean, N. The fractal interpolation for countable systems of data. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 2003, 14, 11–19. [Google Scholar] [CrossRef]
 Secelean, N. Fractal countable interpolation scheme: Existence and affine invariance. Math. Rep. (Bucur.) 2011, 13, 75–87. [Google Scholar]
 Ri, S. A new idea to construct the fractal interpolation function. Indag. Math. 2018, 29, 962–971. [Google Scholar] [CrossRef]
 Kim, J.; Kim, H.; Mun, H. Nonlinear fractal interpolation curves with function vertical scaling factors. Indian J. Pure Appl. Math. 2020, 51, 483–499. [Google Scholar] [CrossRef]
 Ri, S.; Drakopoulos, V. How Are Fractal Interpolation Functions Related to Several Contractions? In Mathematical Theorems—Boundary Value Problems and Approximations; Alexeyeva, L., Ed.; IntechOpen: London, UK, 2020. [Google Scholar]
 Pacurar, C.M. A countable fractal interpolation scheme involving Rakotch contractions. arXiv 2021, arXiv:2102.09855. [Google Scholar]
 Dalla, L.; Drakopoulos, V.; Prodromou, M. On the box dimension for a class of nonaffine fractal interpolation functions. Anal. Theory Appl. 2003, 19, 220–233. [Google Scholar] [CrossRef]
 De Amo, E.; Chiţescu, I.; Diaz Carrillo, M.; Secelean, N.A. A new approximation procedure for fractals. J. Comput. Appl. 2003, 151, 355–370. [Google Scholar] [CrossRef] [Green Version]
 Dubuc, S.; Elqortobi, A. Approximations of fractal sets. J. Comput. Appl. Math. 1990, 29, 79–89. [Google Scholar] [CrossRef] [Green Version]
 Chiţescu, I.; Miculescu, R. Approximation of fractals generated by Fredholm integral equations. J. Comput. Anal. Appl. 2009, 11, 286–293. [Google Scholar]
 Chiţescu, I.; Georgescu, H.; Miculescu, R. Approximation of infinite dimensional fractals generated by integral equations. J. Comput. Appl. Math. 2010, 234, 1417–1425. [Google Scholar] [CrossRef] [Green Version]
 Miculescu, R.; Mihail, A.; Urziceanu, S.A. A new algorithm that generates the image of the attractor of a generalized iterated function system. Numer. Algorithms 2020, 83, 1399–1413. [Google Scholar] [CrossRef] [Green Version]
 Matkowski, J. Integrable solutions of functional equations. Dissertationes Math. 1975, 127, 68. [Google Scholar]
 Hutchinson, J. Fractals and self similarity. Indiana Univ. Math. J. 1981, 30, 713–747. [Google Scholar] [CrossRef]
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Băicoianu, A.; Păcurar, C.M.; Păun, M. A Concretization of an Approximation Method for NonAffine Fractal Interpolation Functions. Mathematics 2021, 9, 767. https://doi.org/10.3390/math9070767
Băicoianu A, Păcurar CM, Păun M. A Concretization of an Approximation Method for NonAffine Fractal Interpolation Functions. Mathematics. 2021; 9(7):767. https://doi.org/10.3390/math9070767
Chicago/Turabian StyleBăicoianu, Alexandra, Cristina Maria Păcurar, and Marius Păun. 2021. "A Concretization of an Approximation Method for NonAffine Fractal Interpolation Functions" Mathematics 9, no. 7: 767. https://doi.org/10.3390/math9070767