Approximation by shape preserving fractal functions with variable scalings

• Published in: Calcolo Vol. 58; no. 1
• Main Authors: Jha, Sangita, Chand, A. K. B., Navascués, M. A.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2021; Springer Nature B.V
• Subjects: Approximation; Chebyshev approximation; Continuity (mathematics); Fractals; Functionals; Interpolation; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical Analysis; Polynomials; Theorems; Theory of Computation; Fractal interpolation functions; Chebyshev system; Müntz approximation; 42A15; Smoothing; 41A30; 28A80; Fractals; Shape preservation; 41A50; 41A15; 41A29
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-021-00396-8

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I . This method produces a class of functions f α ∈ C ( I ) , where α is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the α -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f . We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the α -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the α -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Müntz theorem in C [ 0 , 1 ] .