Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula:: Science Publishing Group

Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula

Ildikó Somogyi, Anna Soós

Department of Mathematics and Computer Science, “Babes¸ Bolyai” University of Cluj-Napoca, Cluj-Napoca, Romania

Applied and Computational Mathematics, Volume 12, Issue 5, October 2023

Pages: 109-113

Received: Jul. 28, 2023

Accepted: Sep. 14, 2023

Published: Sep. 25, 2023

DOI: 10.11648/j.acm.20231205.11

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Abstract

Fractal interpolation methods became an important method in data processing, even for functions with abrupt changes. In the last few decades it has attracted several authors because it can be applied in various fields. The advantage of these methods are that we can generalize the classical approximation methods and also we can combine these methods for example with Lagrange interpolation, Hermite interpolation or spline interpolation. The classical Lagrange interpolation problem give the construction of a suitable approximate function based on the values of the function on given points. These method was generalized for more than one variable functions. In this article we generalize the so-called algebraic maximal Lagrange interpolation formula in order to approximate functions on a rectangular domain with fractal functions. The construction of the fractal function is made with a so-called iterated function system. This method it has the advantage that all classical methods can be obtained as a particular case of a fractal function. We also use the construction for a polynomial type fractal function and we proof that the Lagrange-type algebraic minimal bivariate fractal function satisfies the required interpolation conditions. Also we give a delimitation of the error, using the result regarding the error of a polynomial fractal interpolation function.

Keywords

Fractal Interpolation, Lagrange Interpolation, Fractal Surfaces

To cite this article

Ildikó Somogyi, Anna Soós. (2023). Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula. Applied and Computational Mathematics, 12(5), 109-113. https://doi.org/10.11648/j.acm.20231205.11

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

1.	M. F. Barnsley, Fractal Function and Interpolation, Constr. Approx., 1986, vol. 2, no. 4, pp. 303-329.
2.	M. F. Barnsley, A. N. Harrington, The Calculus of Fractal Interpolation Functions, J. Approx. Theory, 1989, vol. 57, no. 1, pp 14-34.
3.	M. F. Barnsley, M. Hegeland and P. Massopust, Numerics and Fractals, https://arxiv.org/abs/1309.0972, 2014.
4.	A. K. B. Chand, G. P. Kapoor, Generalized Cubic Spline Fractal Interpolation Functions, SIAM Journal on Numerical Analysis, Vol. 44, No. 2, 2006, pp. 655-676.
5.	A. K. B. Chand, M. A. Navascués, Generalized Hermite Fractal Interpolation, Academia de Ciencias, Zaragoza, Vol. 64, 2009, pp. 107-120.
6.	A. K. B. Chand, M. A. Navascués, P. Viswanathan, S. K. Katiyar, Fractal trigonometric polynomial for restricted range approximation Fractals, 24(2), 2016, 1-11.
7.	Gh. Coman, M. Birou, T. Catinas, C. Osan, A. Oprisan, I. Pop, I. Somogyi, I. Todea, Interpolation Operators, Casa Cartii de Stiinte, Cluj, 2004.
8.	L. Dalla, Bivariate Fractal Interpolation Functions on Grids, Fractals, 2002, vol. 10, no. 1, pp. 53-58.
9.	J. E. Hutchinson, L. Rüschendorf L, Selfsimilar Fractals and Selfsimilar Random Fractals, Progress in Probability, vol. 46, 2000, pp. 109-123.
10.	P. R. Massopust, Fractal Surfaces, J. Math. An. Appl, vol. 151, no. 1, 1990, pp. 275-290.
11.	A. M. Navascués, Fractal Polynomial Interpolation, A. M. J. of Analysis and its Approx., vol. 24., No. 2, 2005, pp. 401-418.
12.	M. A. Navascués, S. Jha, A. K. B. Chand, Generalized Bivariate Hermite Fractal Interpolation Function, Numerical Analysis and Applications, vol. 14, no. 2, 2021, pp. 103-114.
13.	M. A. Navascués, J. Sangita, A. K. B. Chand, R. N. Mohapatra, Iterative Schemes Involving Several Mutual Contractions, Mathematics, 11, 2023, pp. 1-18.
14.	I. Somogyi, A. Soós, Graph-directed random fractal interpolation function, Studia Mathematica, Vol 66, Nr 2, 2021, pp. 247-255.
15.	H. Y. Wang, J. S. Yu, Fractal interpolation functions with variable parameters and their analytical properties, J. Approx. Theory 175, 2013, pp. 1-18.