Fractal Interpolation Functions: A Short Survey
Author(s)
María Antonia Navascués*,
Arya Kumar Bedabrata Chand,
Viswanathan Puthan Veedu,
María Victoria Sebastián
Affiliation(s)
Departamento de Matemática Aplicada EINA, Universidad de Zaragoza, Zaragoza, Spain. Department of Mathematics, Indian Institute of Technology, Madras, Chennai, India. Centro Universitario de la Defensa de Zaragoza Academia General Militar, Zaragoza, Spain.
Departamento de Matemática Aplicada EINA, Universidad de Zaragoza, Zaragoza, Spain. Department of Mathematics, Indian Institute of Technology, Madras, Chennai, India. Centro Universitario de la Defensa de Zaragoza Academia General Militar, Zaragoza, Spain.
ABSTRACT
The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.
Cite this paper
Navascués, M. , Chand, A. , Veedu, V. and Sebastián, M. (2014) Fractal Interpolation Functions: A Short Survey. Applied Mathematics, 5, 1834-1841. doi: 10.4236/am.2014.512176.
Navascués, M. , Chand, A. , Veedu, V. and Sebastián, M. (2014) Fractal Interpolation Functions: A Short Survey. Applied Mathematics, 5, 1834-1841. doi: 10.4236/am.2014.512176.
References
[1] Barnsley, M.F. (1986) Fractal Functions and Interpolation. Constructive Approximation, 2, 303-329.
http://dx.doi.org/10.1007/BF01893434 [2] Hutchinson, J.E. (1981) Fractals and Self Similarity. Indiana University Mathematics Journal, 30, 713-747.
http://dx.doi.org/10.1512/iumj.1981.30.30055 [3] Barnsley, M.F. and Harrington, A.N. (1989) The Calculus of Fractal Interpolation Functions. Journal of Approximation Theory, 57, 14-34.
http://dx.doi.org/10.1016/0021-9045(89)90080-4 [4] Navascués, M.A. (2005) Fractal Polynomial Interpolation. Zeitschrift für Analysis und ihre Anwendungen, 25, 401-418. [5] Navascués, M.A. and Chand, A.K.B. (2008) Fundamental Sets of Fractal Functions. Acta Applicandae Mathematicae, 100, 247-261. [6] Navascués, M.A. (2010) Fractal Approximation. Complex Analysis and Operator Theory, 4, 953-974. [7] Navascués, M.A. and Sebastián, M.V. (2013) Numerical Integration of Affine Fractal Functions. Journal of Computational and Applied Mathematics, 252, 169-176. [8] Navascués, M.A. and Sebastián, M.V. (2006) Error Bounds in Affine Fractal Interpolation. Mathematical Inequalities & Applications, 9, 273-288. [9] Navascués, M.A. and Sebastián, M.V. (2007) Construction of Affine Fractal Functions Close to Classical Interpolants. Journal of Computational and Applied Mathematics, 9, 271-283. [10] Navascués, M.A. (2014) Affine Fractal Functions as Bases of Continuous Functions. Quaestiones Mathematicae, 37, 1-14. [11] Chand, A.K.B. and Kapoor, G.P. (2006) Generalized Cubic Spline Fractal Interpolation Functions. SIAM Journal on Numerical Analysis, 44, 655-676.
http://dx.doi.org/10.1137/040611070 [12] Navascués, M.A. and Sebastián, M.V. (2004) Generalization of Hermite Functions by Fractal Interpolation. Journal of Approximation Theory, 131, 19-29. [13] Dalla, L. and Drakopoulos, V. (1999) On the Parameter Identification Problem in the Plane and Polar Fractal Interpolation Functions. Journal of Approximation Theory, 101, 289-302.
http://dx.doi.org/10.1006/jath.1999.3380 [14] Chand, A.K.B. and Viswanathan, P. (2013) A Constructive Approach to Cubic Hermite Fractal Interpolation Function and Its Constrained Aspects. BIT Numerical Mathematics, 53, 841-865.
http://dx.doi.org/10.1007/s10543-013-0442-4 [15] Navascués, M.A. (2007) Non-Smooth Polynomials. International Journal of Analysis and Applications, 1, 159-174. [16] Barnsley, M.F. (1988) Fractals Everywhere. Academic Press, Orlando. [17] Gang, C. (1996) The Smoothness and Dimension of Fractal Interpolation Functions. Applied Mathematics: A Journal of Chinese Universities, 11, 409-418. [18] Wang, H.Y. and Yu, J.S. (2013) Fractal Interpolation Functions with Variable Parameters and Their Analytical Properties. Journal of Approximation Theory, 175, 1-18.
http://dx.doi.org/10.1016/j.jat.2013.07.008 [19] Chand, A.K.B., Vijender, N. and Navascués, M.A. (2014) Shape Preservation of Scientific Data through Rational Fractal Splines. Calcolo, 51, 329-362. [20] Barnsley, M.F., Elton, J., Hardin, D. and Massopust, P. (1989) Hidden Variable Fractal Interpolation Functions. SIAM Journal on Mathematical Analysis, 20, 1218-1242.
http://dx.doi.org/10.1137/0520080 [21] Chand, A.K.B. and Kapoor, G.P. (2008) Stability of Affine Coalescence Hidden Variable Fractal Interpolation Functions. Nonlinear Anal. TMA, 68, 3757-3770. [22] Bouboulis, P. and Dalla, L. (2007) Fractal Interpolation Surfaces Derived from Fractal Interpolation Functions. Journal of Mathematical Analysis and Applications, 336, 919-936.
http://dx.doi.org/10.1016/j.jmaa.2007.01.112 [23] Chand, A.K.B. and Navascués, M.A. (2008) Natural Bicubic Spline Fractal Interpolation. Nonlinear Analysis: Theory, Methods & Applications, 69, 3679-3691. [24] Massopust, P.R. (1990) Fractal Surfaces. Journal of Mathematical Analysis and Applications, 151, 275-290.
http://dx.doi.org/10.1016/0022-247X(90)90257-G [25] Xie, H. and Sun, H. (1997) The Study of Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolation Surfaces. Fractals, 5, 625-634.
http://dx.doi.org/10.1142/S0218348X97000504 [26] Navascués, M.A. and Sebastián, M.V. (2004) Fitting Curves by Fractal Interpolation: An Application to Electroencephalographic Processing. In: Novak, M.M., Ed., Thinking in Patterns: Fractals and Related Phenomena in Nature, World Scientific Publishing, Singapore City, 143-154. [27] Navascués, M.A. and Sebastián, M.V. (2006) Smooth Fractal Interpolation. Journal of Inequalities and Applications, 2006, Article ID: 78734. [28] Viswanathan, P., Chand, A.K.B and Navascués, M.A. (2014) Fractal Perturbation Preserving Fundamental Shapes: Bounds on the Scale Factors. Journal of Mathematical Analysis and Applications, Available Online. [29] Navascués, M.A. (2012) Fractal Bases of Lp Spaces. Fractals, 20, 141-148. [30] Navascués, M.A., Sebastián, M.V. and Valdizán, J.R. (2006) Surface Laplacian and Fractal Brain Mapping. Journal of Computational and Applied Mathematics, 189, 132-141. [31] Navascués, M.A. and Sebastián, M.V. (2006) Spectral and Affine Fractal Methods in Signal Processing. International Mathematical Forum, 1, 1405-1422. [32] Navascués, M.A. and Sebastián, M.V. (2012) Legendre Transform of Sampled Signals by Fractal Methods. Monografías Seminario Matemático García de Galdeano, 37, 181-188.
[1] Barnsley, M.F. (1986) Fractal Functions and Interpolation. Constructive Approximation, 2, 303-329.
http://dx.doi.org/10.1007/BF01893434 [2] Hutchinson, J.E. (1981) Fractals and Self Similarity. Indiana University Mathematics Journal, 30, 713-747.
http://dx.doi.org/10.1512/iumj.1981.30.30055 [3] Barnsley, M.F. and Harrington, A.N. (1989) The Calculus of Fractal Interpolation Functions. Journal of Approximation Theory, 57, 14-34.
http://dx.doi.org/10.1016/0021-9045(89)90080-4 [4] Navascués, M.A. (2005) Fractal Polynomial Interpolation. Zeitschrift für Analysis und ihre Anwendungen, 25, 401-418. [5] Navascués, M.A. and Chand, A.K.B. (2008) Fundamental Sets of Fractal Functions. Acta Applicandae Mathematicae, 100, 247-261. [6] Navascués, M.A. (2010) Fractal Approximation. Complex Analysis and Operator Theory, 4, 953-974. [7] Navascués, M.A. and Sebastián, M.V. (2013) Numerical Integration of Affine Fractal Functions. Journal of Computational and Applied Mathematics, 252, 169-176. [8] Navascués, M.A. and Sebastián, M.V. (2006) Error Bounds in Affine Fractal Interpolation. Mathematical Inequalities & Applications, 9, 273-288. [9] Navascués, M.A. and Sebastián, M.V. (2007) Construction of Affine Fractal Functions Close to Classical Interpolants. Journal of Computational and Applied Mathematics, 9, 271-283. [10] Navascués, M.A. (2014) Affine Fractal Functions as Bases of Continuous Functions. Quaestiones Mathematicae, 37, 1-14. [11] Chand, A.K.B. and Kapoor, G.P. (2006) Generalized Cubic Spline Fractal Interpolation Functions. SIAM Journal on Numerical Analysis, 44, 655-676.
http://dx.doi.org/10.1137/040611070 [12] Navascués, M.A. and Sebastián, M.V. (2004) Generalization of Hermite Functions by Fractal Interpolation. Journal of Approximation Theory, 131, 19-29. [13] Dalla, L. and Drakopoulos, V. (1999) On the Parameter Identification Problem in the Plane and Polar Fractal Interpolation Functions. Journal of Approximation Theory, 101, 289-302.
http://dx.doi.org/10.1006/jath.1999.3380 [14] Chand, A.K.B. and Viswanathan, P. (2013) A Constructive Approach to Cubic Hermite Fractal Interpolation Function and Its Constrained Aspects. BIT Numerical Mathematics, 53, 841-865.
http://dx.doi.org/10.1007/s10543-013-0442-4 [15] Navascués, M.A. (2007) Non-Smooth Polynomials. International Journal of Analysis and Applications, 1, 159-174. [16] Barnsley, M.F. (1988) Fractals Everywhere. Academic Press, Orlando. [17] Gang, C. (1996) The Smoothness and Dimension of Fractal Interpolation Functions. Applied Mathematics: A Journal of Chinese Universities, 11, 409-418. [18] Wang, H.Y. and Yu, J.S. (2013) Fractal Interpolation Functions with Variable Parameters and Their Analytical Properties. Journal of Approximation Theory, 175, 1-18.
http://dx.doi.org/10.1016/j.jat.2013.07.008 [19] Chand, A.K.B., Vijender, N. and Navascués, M.A. (2014) Shape Preservation of Scientific Data through Rational Fractal Splines. Calcolo, 51, 329-362. [20] Barnsley, M.F., Elton, J., Hardin, D. and Massopust, P. (1989) Hidden Variable Fractal Interpolation Functions. SIAM Journal on Mathematical Analysis, 20, 1218-1242.
http://dx.doi.org/10.1137/0520080 [21] Chand, A.K.B. and Kapoor, G.P. (2008) Stability of Affine Coalescence Hidden Variable Fractal Interpolation Functions. Nonlinear Anal. TMA, 68, 3757-3770. [22] Bouboulis, P. and Dalla, L. (2007) Fractal Interpolation Surfaces Derived from Fractal Interpolation Functions. Journal of Mathematical Analysis and Applications, 336, 919-936.
http://dx.doi.org/10.1016/j.jmaa.2007.01.112 [23] Chand, A.K.B. and Navascués, M.A. (2008) Natural Bicubic Spline Fractal Interpolation. Nonlinear Analysis: Theory, Methods & Applications, 69, 3679-3691. [24] Massopust, P.R. (1990) Fractal Surfaces. Journal of Mathematical Analysis and Applications, 151, 275-290.
http://dx.doi.org/10.1016/0022-247X(90)90257-G [25] Xie, H. and Sun, H. (1997) The Study of Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolation Surfaces. Fractals, 5, 625-634.
http://dx.doi.org/10.1142/S0218348X97000504 [26] Navascués, M.A. and Sebastián, M.V. (2004) Fitting Curves by Fractal Interpolation: An Application to Electroencephalographic Processing. In: Novak, M.M., Ed., Thinking in Patterns: Fractals and Related Phenomena in Nature, World Scientific Publishing, Singapore City, 143-154. [27] Navascués, M.A. and Sebastián, M.V. (2006) Smooth Fractal Interpolation. Journal of Inequalities and Applications, 2006, Article ID: 78734. [28] Viswanathan, P., Chand, A.K.B and Navascués, M.A. (2014) Fractal Perturbation Preserving Fundamental Shapes: Bounds on the Scale Factors. Journal of Mathematical Analysis and Applications, Available Online. [29] Navascués, M.A. (2012) Fractal Bases of Lp Spaces. Fractals, 20, 141-148. [30] Navascués, M.A., Sebastián, M.V. and Valdizán, J.R. (2006) Surface Laplacian and Fractal Brain Mapping. Journal of Computational and Applied Mathematics, 189, 132-141. [31] Navascués, M.A. and Sebastián, M.V. (2006) Spectral and Affine Fractal Methods in Signal Processing. International Mathematical Forum, 1, 1405-1422. [32] Navascués, M.A. and Sebastián, M.V. (2012) Legendre Transform of Sampled Signals by Fractal Methods. Monografías Seminario Matemático García de Galdeano, 37, 181-188.