×

Spline coalescence hidden variable fractal interpolation functions. (English) Zbl 1146.41003

Summary: This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of M. F. Barnsley and A. N. Harrington [J. Approximation Theory 57, 14–34 (1989; Zbl 0693.41008)], when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.

MSC:

41A15 Spline approximation
28A80 Fractals
65D07 Numerical computation using splines

Citations:

Zbl 0693.41008
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] M. F. Barnsley, “Fractal functions and interpolation,” Constructive Approximation, vol. 2, no. 4, pp. 303-329, 1986. · Zbl 0606.41005 · doi:10.1007/BF01893434
[2] M. F. Barnsley, Fractals Everywhere, Academic Press, Massachusetts, 1988. · Zbl 0691.58001
[3] M. F. Barnsley, J. Elton, D. Hardin, and P. R. Massopust, “Hidden variable fractal interpolation functions,” SIAM Journal on Mathematical Analysis, vol. 20, no. 5, pp. 1218-1242, 1989. · Zbl 0704.26009 · doi:10.1137/0520080
[4] M. F. Barnsley and A. N. Harrington, “The calculus of fractal interpolation functions,” Journal of Approximation Theory, vol. 57, no. 1, pp. 14-34, 1989. · Zbl 0693.41008 · doi:10.1016/0021-9045(89)90080-4
[5] M. F. Barnsley and L. P. Hurd, Fractal Image Compression, A K Peters, Massachusetts, 1993. · Zbl 0796.68186
[6] A. K. B. Chand, A study on coalescence and spline fractal interpolation function, M.S. thesis, IIT Kanpur, Kanpur, 2005.
[7] A. K. B. Chand and G. P. Kapoor, “Hidden variable bivariate fractal interpolation surfaces,” Fractals, vol. 11, no. 3, pp. 277-288, 2003. · Zbl 1046.28004 · doi:10.1142/S0218348X03002129
[8] A. K. B. Chand and G. P. Kapoor, “Generalized cubic spline fractal interpolation functions,” SIAM Journal on Numerical Analysis, vol. 44, no. 2, pp. 655-676, 2006. · Zbl 1136.41006 · doi:10.1137/040611070
[9] A. K. B. Chand and G. P. Kapoor, “Coalescence hidden variable fractal interpolation functions and its smoothness analysis,” preprint, 2005, http://arxiv.org/abs/math.DS/0511073. · Zbl 1146.41003
[10] H. H. Hardy and R. A. Beier, Fractals in Reservoir Engineering, World Scientific, Singapore, 1994.
[11] J. E. Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics Journal, vol. 30, no. 5, pp. 713-747, 1981. · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[12] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, California, 1977. · Zbl 0504.28001
[13] P. Maragos, “Fractal aspects of speech signals: dimension and interpolation,” in International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’91), vol. 1, pp. 417-420, Ontario, 1991.
[14] P. R. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, California, 1994. · Zbl 0817.28004
[15] D. S. Mazel and M. H. Hayes, “Using iterated function systems to model discrete sequences,” IEEE Transactions on Signal Processing, vol. 40, no. 7, pp. 1724-1734, 1992. · Zbl 0753.58015 · doi:10.1109/78.143444
[16] M. A. Navascués, “Fractal polynomial interpolation,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 2, pp. 401-418, 2005. · Zbl 1082.28006 · doi:10.4171/ZAA/1248
[17] M. A. Navascués and M. V. Sebastián, “Some results of convergence of cubic spline fractal interpolation functions,” Fractals, vol. 11, no. 1, pp. 1-7, 2003. · Zbl 1053.41013 · doi:10.1142/S0218348X03001550
[18] M. A. Navascués and M. V. Sebastián, “Generalization of Hermite functions by fractal interpolation,” Journal of Approximation Theory, vol. 131, no. 1, pp. 19-29, 2004. · Zbl 1068.41006 · doi:10.1016/j.jat.2004.09.001
[19] M. A. Navascués and M. V. Sebastián, “Smooth fractal interpolation,” Journal of Inequalities and Applications,, vol. 2006, Article ID 78734, p. 20, 2006. · Zbl 1133.41307 · doi:10.1155/JIA/2006/78734
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.