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Chand, Arya K. B.; Katiyar, Saurabh K.; Viswanathan, Puthan V.

Approximation using hidden variable fractal interpolation function. (English) Zbl 1318.28017

J. Fractal Geom. 2, No. 1, 81-114 (2015).
Given a vector-valued function \(f:I\to \mathbb{R}^2\), where \(I\) denotes a compact interval in \(\mathbb{R}\), and a family of upper triangular \(2\times 2\)-matrix \(\{A\}_{n = 0}^{N-1}\), \(1 < N\in \mathbb{N}\), the authors associate with the pair \((f,A)\) hidden variable fractal interpolation functions denoted by \(f[A]\). The entries in the matrices \(A_n\) determine the regularity properties of the ensuing hidden variable fractal function \(f[A]\). Estimates for the approximation error between \(f\) and \(f[A]\) are derived and suitable values for the entries in the matrices \(A_n\) identified to ensure the preservation of monotonicity and \(C^1\)-continuity of \(f\).
Reviewer: Peter Massopust (München)

Cited in 12 Documents

MSC:

28A80 Fractals
41A05 Interpolation in approximation theory
41A10 Approximation by polynomials

Keywords:

fractal operator; fractal cubic spline; hidden variable fractal interpolation function; convex optimization; sensitivity analysis; positivity; parameter identification
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References:

[1] G. Bachman and L. Narici, Functional analysis. Reprint of the 1966 original. Dover Publications, Mineola, N.Y., 2000. · Zbl 0983.46001
[2] M. F. Barnsley, Fractals everywhere. Academic Press, Inc., Boston, MA, etc., 1988. 2nd ed. revised with the assistance of and with a foreword by Hawley Rising, III. Academic Press Professional, Boston, MA, 1993. 3rd revised ed., Dover Pub- lications, Mineola, N.Y., 2012. · Zbl 0691.58001
[3] M. F. Barnsley, Fractal functions and interpolation. Constr. Approx. 2 (1986), no 4, 303-329. · Zbl 0606.41005 · doi:10.1007/BF01893434
[4] M. F. Barnsley and A. N. Harrington, The calculus of fractal interpolation functions. J. Approx. Theory 57 (1989), no. 1, 14-34. · Zbl 0693.41008 · doi:10.1016/0021-9045(89)90080-4
[5] M. F. Barnsley, J. Elton, D. Hardin, and P. Massopust, Hidden variable fractal inter- polation functions. SIAM J. Math. Anal. 20 (1989), no. 5, 1218-1242. · Zbl 0704.26009 · doi:10.1137/0520080
[6] B. Bollabás, Linear analysis. An introductory course. 2nd ed. Cambridge Math- ematical Textbooks Cambridge University Press, Cambridge, 1999. · Zbl 0923.46002
[7] S. Butt and K. W. Brodlie, Preserving positivity using piecewise cubic interpolation. Comput. &amp; Graphics 17 (1993), no. 1, 55-64.
[8] A. K. B. Chand and G. P. Kapoor, Generalized cubic spline fractal interpola- tion functions. SIAM J. Numer. Anal. 44 (2006), no. 2, 655-676. · Zbl 1136.41006 · doi:10.1137/040611070
[9] A. K. B. Chand and G. P. Kapoor, Stability of affine coalescence hidden vari- able fractal interpolation functions. Nonlinear Anal. 68 (2008), no. 12, 3757-3770. · Zbl 1147.26004 · doi:10.1016/j.na.2007.04.017
[10] A. K. B. Chand and P. Viswanathan, A constructive approach to cubic Hermite frac- tal interpolation function and its constrained aspects. BIT 53 (2013), no. 4, 841-865. · Zbl 1283.65010 · doi:10.1007/s10543-013-0442-4
[11] P. Costantini, Curve and surface design using variable degree polynomial splines. Comput Aided Geom Des. 17 (2000), no. 5, 426-446. · Zbl 0938.68128 · doi:10.1016/S0167-8396(00)00010-8
[12] L. Dalla and V. Drakopoulos, On the parameter identification problem in the plane and the polar fractal interpolation. J. Approx. Theory 101 (1999), 289-302. · Zbl 0945.41001 · doi:10.1006/jath.1999.3380
[13] J. A. Gregory and R. Delbourgo, Shape preserving piecewise rational interpolation. SIAM J. Stat. Comput. 6 (1985), no. 4, 967-976. · Zbl 0586.65006 · doi:10.1137/0906065
[14] X. Han, Convexity-preserving piecewise rational quartic interpolation. SIAM. J. Nu- mer. Anal. 46 (2008), no. 2, 920-929. · Zbl 1165.65005 · doi:10.1137/060671577
[15] J. Lévy-Véhel and E. Lutton, Optimization of fractal functions using genetic algo- rithms. INRIA, Rapport de reserche no. 1941, 1993.
[16] P. Manousopoulos, V. Drakopoulos, and T. Theoharis, Parameter identification of 1D fractal interpolation functions using bounding volumes. J. Comput. Appl. Math. 233 (2009), no. 4, 1063-1082. · Zbl 1177.65024 · doi:10.1016/j.cam.2009.08.115
[17] M. A. Marvasti and W. Strahle, Fractal geometry analysis of turbulent data. Signal Processing 41 (1995), no. 2, 191-201. · Zbl 0880.76034 · doi:10.1016/0165-1684(94)00100-E
[18] P. Massopust, Fractal functions, fractal surfaces, and wavelets. Academic Press, San Diego, CA, 1994. · Zbl 0817.28004
[19] M. A. Navascués, Fractal polynomial interpolation. Z. Anal. Anwendungen 24 (2005), no. 2, 401-418. · Zbl 1082.28006 · doi:10.4171/ZAA/1248
[20] M. A. Navascués, Fractal approximation. Complex Anal. Oper. Theory 4 (2010), no. 4, 953-974. · Zbl 1202.28013 · doi:10.1007/s11785-009-0033-1
[21] M. A. Navascués and A. K. B. Chand, Fundamental sets of fractal functions. Acta Appl. Math. 100 (2008), no. 3, 247-261. · Zbl 1132.28005 · doi:10.1007/s10440-007-9182-2
[22] M. A. Navascués and M. V. Sebastián, Construction of affine fractal functions close to classical interpolants. J. Comp. Anal. Appl. 9 (2007), no. 3, 271-285. · Zbl 1128.28008
[23] H. Raun, J. Z. Sha, and W. Y. Su, Counterexamples in parameter identification prob- lem of the fractal interpolation functions. J. Approx. Theory 122 (2003), no. 1, 121-128. · Zbl 1024.41001 · doi:10.1016/S0021-9045(03)00028-5
[24] J. W. Schimdt and W. Heß, Positivity of cubic polynomial on intervals and positive spline interpolation. BIT 28 (1988), no. 2, 340-352. · Zbl 0642.41007 · doi:10.1007/BF01934097
[25] P. Viswanathan and A. K. B. Chand, Fractal rational functions and their approxima- tion properties. J. Approx. Theory 185 (2014), 31-50. · Zbl 1304.41014 · doi:10.1016/j.jat.2014.05.013
[26] P. Viswanathan and A. K. B. Chand, A fractal procedure for monotonicity preserving interpolation. Appl. Math. Comput. 247 (2014), 190-204. · Zbl 1338.65030
[27] P. Viswanathan and A. K. B. Chand, \?-fractal rational splines for constrained inter- polation. Elect. Tran. Numer. Anal. 41 (2014), 420-442.
[28] P. Viswanathan and A. K. B. Chand, On cubic Hermite coalescence hidden variable fractal interpolation functions. Appl. Math. J. Chinese Univ. Ser. B 30 (2015), no. 1, 55-76. · Zbl 1340.41003
[29] H. Y. Wang and J. S. Yu, Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175 (2013), 1-18. · Zbl 1303.28014 · doi:10.1016/j.jat.2013.07.008
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