Bivariate shape preserving interpolation: A fractal-classical hybrid approach. (December 2015)

• Record Type: Journal Article
• Authors: Chand, A.K.B.; Viswanathan, P.; Vijender, N.
• Is Part Of: Chaos, solitons and fractals. Volume 81:Part A(2015)
• Page Start: 330
• Page End: 344
• Publication Date: 2015-12
• Subjects: Fractal interpolation -- Blending functions -- Bicubic partially blended fractal surface -- Convergence -- Positivity -- Monotonicity; 28A80 -- 26C15 -- 41A05 -- 41A29 -- 65D05; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2015.10.002
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: The notion of cubic fractal interpolation function (FIF) has received considerable attention in the literature due to its versatility, flexibility and ease of implementation. In this article, we shall view cubic FIFs as a family of C 1 -continuous fractal functions associated with the traditional C 1 -continuous cubic spline. General theorems that identify suitable values of the parameters so as to constrain a fractal function and its first derivative within suitable axis-aligned rectangles are reported. By applying these theorems, cubic fractal interpolation of a data set subject to strip conditions on the interpolant and its first derivative is discussed. These results are applied to investigate positivity and monotonicity properties of a hybrid bivariate interpolant over a rectangular region R obtained by blending univariate cubic FIFs via bicubically blended Coons patch. The L ∞ -norm of the error in approximating a function f ∈ C 2 ( R ) with the proposed bivariate interpolant is shown to be of order O ( h 2 ) as h → 0.