A rational iterated function system for resolution of univariate constrained interpolation

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 109; no. 2; pp. 483 - 509
• Main Authors: Viswanathan, P., Chand, A. K. B., Navascués, M. A.
• Format: Journal Article
• Language: English
• Published: Milan Springer Milan 01.09.2015; Springer Nature B.V
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Theoretical; 41A30; Rational quadratic fractal interpolation function; Iterated function systems; Positivity; 28A80; 65D05; 26C15; Parameter identification; Monotonicity; 65D07; Convergence
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-014-0197-z

Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function Φ ∈ C 3 [ x 1 , x n ] with h 3 order of convergence, where h denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.