On Complete Bicubic Fractal Splines

ABSTRACT

Fractal geometry provides a new insight to the approximation and modelling of experimental data. We give the construction of complete cubic fractal splines from a suitable basis and their error bounds with the original function. These univariate properties are then used to investigate complete bicubic fractal splines over a rectangle Bicubic fractal splines are invariant in all scales and they generalize classical bicubic splines. Finally, for an original function , upper bounds of the error for the complete bicubic fractal splines and derivatives are deduced. The effect of equal and non-equal scaling vectors on complete bicubic fractal splines were illustrated with suitably chosen examples.

Fractal geometry provides a new insight to the approximation and modelling of experimental data. We give the construction of complete cubic fractal splines from a suitable basis and their error bounds with the original function. These univariate properties are then used to investigate complete bicubic fractal splines over a rectangle Bicubic fractal splines are invariant in all scales and they generalize classical bicubic splines. Finally, for an original function , upper bounds of the error for the complete bicubic fractal splines and derivatives are deduced. The effect of equal and non-equal scaling vectors on complete bicubic fractal splines were illustrated with suitably chosen examples.

KEYWORDS

Fractals, Iterated Function Systems, Fractal Interpolation Functions, Fractal Splines, Surface Approximation

Fractals, Iterated Function Systems, Fractal Interpolation Functions, Fractal Splines, Surface Approximation

Cite this paper

nullA. Chand and M. Navascués, "On Complete Bicubic Fractal Splines,"*Applied Mathematics*, Vol. 1 No. 3, 2010, pp. 200-210. doi: 10.4236/am.2010.13024.

nullA. Chand and M. Navascués, "On Complete Bicubic Fractal Splines,"

References

[1] I. J. Schoenberg, “Contribution to the Problem of Approximation of Equidistant Data by Analytic Functions, Part A and B,” Quarterly of Applied Mathematics, Vol. 4, No. 2, 1946, pp. 45-99, 112-141.

[2] M. F. Barnsley, “Fractals Everywhere,” Academic Press, Orlando, Florida, 1988.

[3] M. F. Barnsley, “Fractal Functions and Interpolation,” Constructive Approximation, Vol. 2, No. 2, 1986, pp. 303-329.

[4] D. S. Mazel and M. H. Hayes, “Using Iterated Function Systems to Model Discrete Sequences,” IEEE Transactions on Signal Processing, Vol. 40, No. 7, 1992, pp. 1724-1734.

[5] M. F. Barnsley and A. N. Harrington, “The Calculus of Fractal Interpolation Functions,” Journal of Approximation Theory, Vol. 57, No. 1, 1989, pp. 14-34.

[6] A. K. B. Chand and G. P. Kapoor, “Generalized Cubic Spline Fractal Interpolation Functions,” SIAM Journal on Numerical Analysis, Vol. 44, No. 2, 2006, pp. 655-676.

[7] M. A. Navascués and M. V. Sebastián, “Smooth Fractal Interpolation,” Journal of Inequalities and Applications, 2006, pp. 1-20.

[8] M. A. Navascués, “Fractal Polynomial Interpolation,” Zeitschrift fur Analysis und ihre Anwendungen, Vol. 25, No. 2, 2005, pp. 401-418.

[9] M. A. Navascués, “A Fractal Approximation to Periodicity,” Fractals, Vol. 14, No. 4, 2006, pp. 315-325.

[10] A. K. B. Chand and M. A. Navascués, “Generalized Hermite Fractal Interpolation,” Academia de Ciencias, Zara-goza, Vol. 64, 2009, pp. 107-120.

[11] P. R. Massopust, “Fractal surfaces,” Journal of Mathematical Analysis and Applications, Vol. 151, No. 1, 1990, pp. 275-290.

[12] J. S. Geronimo and D. Hardin, “Fractal Interpolation Functions from and their Projections,” Zeit- schrift fur Analysis und ihre Anwendungen, Vol. 12, No. 3, 1993, pp. 535-548.

[13] N. Zhao, “Construction and Application of Fractal Interpolation Surfaces,” Visual Computer, Vol. 12, No. 3, 1996, pp. 132-146.

[14] H. Xie and H. Sun, “The Study of Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolation Surfaces,” Fractals, Vol. 5, No. 4, 1997, pp. 625- 634.

[15] A. K. B. Chand and G. P. Kapoor, “Hidden Variable Bivariate Fractal Interpolation Surfaces,” Fractals, Vol. 11, No. 3, 2003, pp. 277-288.

[16] P. Bouboulis and L. Dalla, “Fractal Interpolation Surfaces derived from Fractal Interpolation Functions,” Journal of Mathematical Analysis and Applications, Vol. 336, No. 2, 2007, pp. 919-936.

[17] R. E. Carlson and C. A. Hall, “Error Bounds for Bicubic Spline Interpolation,” Journal of Approximation Theory, Vol. 7, 1973, pp. 41-47.

[18] C. A. Hall and W. W. Meyer, “Optimal Error Bounds for Cubic Spline Interpolation,” Journal of Approximation Theory, Vol. 16, No. 2, 1976, pp. 105-122.

[19] C. de Boor, “Bicubic Spline Interpolation,” Journal of Mathematics and Physics, Vol. 41, No. 4, 1962, pp. 212- 218.

[20] J. H. Ahlberg, E. N. Nilson and J. L. Walsh, “The Theory of Splines and their Applications,” Academic Press, New York, 1967.

[1] I. J. Schoenberg, “Contribution to the Problem of Approximation of Equidistant Data by Analytic Functions, Part A and B,” Quarterly of Applied Mathematics, Vol. 4, No. 2, 1946, pp. 45-99, 112-141.

[2] M. F. Barnsley, “Fractals Everywhere,” Academic Press, Orlando, Florida, 1988.

[3] M. F. Barnsley, “Fractal Functions and Interpolation,” Constructive Approximation, Vol. 2, No. 2, 1986, pp. 303-329.

[4] D. S. Mazel and M. H. Hayes, “Using Iterated Function Systems to Model Discrete Sequences,” IEEE Transactions on Signal Processing, Vol. 40, No. 7, 1992, pp. 1724-1734.

[5] M. F. Barnsley and A. N. Harrington, “The Calculus of Fractal Interpolation Functions,” Journal of Approximation Theory, Vol. 57, No. 1, 1989, pp. 14-34.

[6] A. K. B. Chand and G. P. Kapoor, “Generalized Cubic Spline Fractal Interpolation Functions,” SIAM Journal on Numerical Analysis, Vol. 44, No. 2, 2006, pp. 655-676.

[7] M. A. Navascués and M. V. Sebastián, “Smooth Fractal Interpolation,” Journal of Inequalities and Applications, 2006, pp. 1-20.

[8] M. A. Navascués, “Fractal Polynomial Interpolation,” Zeitschrift fur Analysis und ihre Anwendungen, Vol. 25, No. 2, 2005, pp. 401-418.

[9] M. A. Navascués, “A Fractal Approximation to Periodicity,” Fractals, Vol. 14, No. 4, 2006, pp. 315-325.

[10] A. K. B. Chand and M. A. Navascués, “Generalized Hermite Fractal Interpolation,” Academia de Ciencias, Zara-goza, Vol. 64, 2009, pp. 107-120.

[11] P. R. Massopust, “Fractal surfaces,” Journal of Mathematical Analysis and Applications, Vol. 151, No. 1, 1990, pp. 275-290.

[12] J. S. Geronimo and D. Hardin, “Fractal Interpolation Functions from and their Projections,” Zeit- schrift fur Analysis und ihre Anwendungen, Vol. 12, No. 3, 1993, pp. 535-548.

[13] N. Zhao, “Construction and Application of Fractal Interpolation Surfaces,” Visual Computer, Vol. 12, No. 3, 1996, pp. 132-146.

[14] H. Xie and H. Sun, “The Study of Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolation Surfaces,” Fractals, Vol. 5, No. 4, 1997, pp. 625- 634.

[15] A. K. B. Chand and G. P. Kapoor, “Hidden Variable Bivariate Fractal Interpolation Surfaces,” Fractals, Vol. 11, No. 3, 2003, pp. 277-288.

[16] P. Bouboulis and L. Dalla, “Fractal Interpolation Surfaces derived from Fractal Interpolation Functions,” Journal of Mathematical Analysis and Applications, Vol. 336, No. 2, 2007, pp. 919-936.

[17] R. E. Carlson and C. A. Hall, “Error Bounds for Bicubic Spline Interpolation,” Journal of Approximation Theory, Vol. 7, 1973, pp. 41-47.

[18] C. A. Hall and W. W. Meyer, “Optimal Error Bounds for Cubic Spline Interpolation,” Journal of Approximation Theory, Vol. 16, No. 2, 1976, pp. 105-122.

[19] C. de Boor, “Bicubic Spline Interpolation,” Journal of Mathematics and Physics, Vol. 41, No. 4, 1962, pp. 212- 218.

[20] J. H. Ahlberg, E. N. Nilson and J. L. Walsh, “The Theory of Splines and their Applications,” Academic Press, New York, 1967.