Constrained and convex interpolation through rational cubic fractal interpolation surface
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Constrained and convex interpolation through rational cubic fractal interpolation surface N. Balasubramani1
· M. Guru Prem Prasad1 · S. Natesan1
Received: 22 September 2017 / Revised: 31 July 2018 / Accepted: 2 August 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract A new C 1 -rational cubic fractal interpolation surface is introduced to interpolate the surface data which lies on a rectangular grid. At the beginning, C 1 -rational cubic fractal interpolation functions are constructed along the grid lines. Then, the C 1 -rational cubic fractal interpolation surface is constructed with the help of blending functions and the rational cubic fractal interpolation functions. Convergence analysis of the fractal interpolation surface to an original function is carried out. Further, the scaling factors and the shape parameters are constrained, so that the fractal interpolation surface would lie above the fixed plane whenever the surface data lie above the plane. Also, conditions on the scaling factors and the shape parameters are derived to interpolate the convex surface data. Keywords Fractal interpolation surface · Convergence · Constrained interpolation · Convexity Mathematics Subject Classification 28A80 · 26C15 · 41A05 · 41A29
1 Introduction Fractal interpolation is an eminent tool to interpolate experimental and/or geometric data. Fractal interpolation is useful in many areas, for example, in tumor perfusion (Craciunescu et al. 2001), neural networks (Severyanov 1997), hydrology (Puente 1996), etc. Iterated function system (IFS) (Hutchinson 1981) is a fundamental object in constructing the fractal interpolation functions (FIFs). Barnsley (1986) constructed the FIF to interpolate the univari-
Communicated by Armin Iske.
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N. Balasubramani [email protected] M. Guru Prem Prasad [email protected] S. Natesan [email protected]
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Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India
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N. Balasubramani et al.
ate data with the help of the IFS that satisfies certain conditions. FIF has a set of free variables called scaling factors and by varying these scaling factors, we can generate a wide variety of FIFs for the same interpolation data. Hence, it is not unique which is contrast to the unique representation in traditional interpolation methods like polynomial, spline, etc. FIFs need not be differentiable and hence fractal interpolation provides a method to approximate a function that is not differentiable. To construct the smooth FIFs, by restricting the scaling factors, Barnsley and Harrington (1989) derived the conditions for smooth FIFs. Owing to Barnsley and Harrington results, various traditional interpolation methods (polynomial, spline, etc.) are generalized in Chand and Viswanathan (2013), Viswanathan et al. (2014), Chand et al. (2014), Navascués and Sebastián (2004, 2006), Navascués (2005), Balasubramani (2017). Massopust (1990) introduced the concept of fractal interpolation surface (FIS) on the trian
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