Abstract
This paper investigates some univariate and bivariate constrained interpolation problems using fractal interpolation functions. First, we obtain rational cubic fractal interpolation functions lying above a prescribed straight line. Using a transfinite interpolation via blending functions, we extend the properties of the univariate rational cubic fractal interpolation function to generate surfaces that lie above a plane. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. Uniform convergence of the bivariate fractal interpolant to the original function which generates the data is proven.
Similar content being viewed by others
References
Barnsley MF (1986) Fractal functions and interpolation. Constr Approx 2:303–329
Barnsley MF, Harrington AN (1989) The calculus of fractal interpolation functions. J Approx Theory 57(1):14–34
Bouboulis P, Dalla L (2007) Fractal interpolation surfaces derived from fractal interpolation functions. J Math Anal Appl 336:919–936
Bouboulis P, Dalla L, Drakopoulos V (2006) Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension. J Approx Theory 141:99–117
Casciola G, Romani L (2003) Rational interpolants with tension parameters. In: Cohen A, Merrien JL, Schumaker LL (eds) Curve and surface design. Nashboro, Brentwood, p 4150
Chand AKB, Kapoor GP (2006) Generalized cubic spline fractal interpolation functions. SIAM J Numer Anal 44(2):655–676
Chand AKB, Kapoor GP (2003) Hidden variable bivariate fractal interpolation surfaces. Fractals 11(3):277–288
Chand AKB, Vijender N, Agarwal RP (2014) Rational iterated function system for positive/monotonic shape preservation. Adv Differ Equ 30:20
Chand AKB, Vijender N, Navascués MA (2013) Shape preservation of scientific data through rational fractal splines. Calcolo 51:329–362
Chand AKB, Viswanathan P (2013) A constructive approach to cubic Hermite fractal interpolation function and its constrained aspects. BIT Numer Math 53:841–865
Dalla L, Drakopoulos V (1999) On the parameter identification problem in the plane and the polar fractal interpolation functions. J Approx Theory 101:289–302
Dalla L (2002) Bivariate fractal interpolation function on grids. Fractals 10(1):53–58
Farin G (2002) Curves and surfaces for CAGD. Morgan Kaufmann, Burlington
Feng Z, Feng Y, Yuan Z (2012) Fractal interpolation surfaces with function vertical scaling factors. Appl Math Lett 25(11):1896–1900
Fritsch FN, Butland J (1984) A method for constructing local monotone piecewise cubic interpolants. SIAM J Sci Stat Comput 5:300–304
Hussain MZ, Hussain M (2006) Visualization of data subject to positive constraints. J Inform Comput Sci 1(3):149–160
Malysz R (2006) The Minkowski dimension of the bivariate fractal interpolation surfaces. Chaos Solitons Fractals 27(5):1147–1156
Massopust P (1997) Fractal functions and their applications. Chaos Solitons Fractals 8(2):171–190
Massopust P (1990) Fractal surfaces. J Math Anal Appl 151:275–290
Metzler W, Yun CH (2010) Construction of fractal interpolation surfaces on rectangular grids. Int J Bifurcat Chaos 20:4079–4086
Navascués MA (2005) Fractal polynomial interpolation. Z Anal Anwend 24(2):1–20
Navascués MA, Sebastián MV (2004) Generalization of Hermite functions by fractal interpolation. J Approx Theory 131(1):19–29
Navascués MA, Sebastián MV (2006) Smooth fractal interpolation. J Inequal Appl Article ID 78734:1–20
Navascués MA, Chand AKB, Viswanathan P, Sebastián MV (2014) Fractal interpolation functions: a short survey. Appl Math 5:1834–1841
Schmidt JW, Heß W (1988) Positivity of cubic polynomials on intervals and positive spline interpolation, BIT. Numer Math 28:340–352
Viswanathan P, Chand AKB (2013) A new class of rational cubic fractal splines for univariate interpolation. In: Mohapatra RN, Giri D, Saxena PK, Srivastava PD (eds) Mathematics and computing 2013. Springer proceedings in mathematics & statistics, vol 91. Springer, India, pp 103–120
Viswanathan P, Chand AKB (2014) A fractal procedure for monotonicity preserving interpolation. Appl Math Comput 247:190–204
Viswanathan P, Chand AKB (2015) A \({\cal {C}}^{1}\) rational cubic fractal interpolation function: convergence and associated parameter identification problem. Acta Appl Math 136(1):262–276
Viswanathan P, Chand AKB, Navascués MA (2014) Fractal perturbation preserving fundamental shapes: Bounds on the scale factors. J Math Anal Appl 419:804–817
Xie H, Sun H (1997) The study of bivariate fractal interpolation functions and creation of fractal interpolated surfaces. Fractals 5(4):625–34
Acknowledgments
The first two authors wish to thank the Science and Engineering Research Council (SERC), Department of Science and Technology (DST) India (Project No. SR/S4/MS: 694/10).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Antonio José Silva Neto.
Rights and permissions
About this article
Cite this article
Chand, A.K.B., Viswanathan, P. & Vijender, N. Bicubic partially blended rational fractal surface for a constrained interpolation problem. Comp. Appl. Math. 37, 785–804 (2018). https://doi.org/10.1007/s40314-016-0373-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-016-0373-1
Keywords
- Blending functions
- Fractal interpolation
- Bicubic partially blended fractal surface
- Convergence
- Constrained interpolation
- Positivity