Approximation by shape preserving fractal functions with variable scalings


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 \(f^{\varvec{\alpha }}\in {\mathcal {C}}(I)\), where \(\varvec{\alpha }\) 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 \(\varvec{\alpha }\)-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 \(\varvec{\alpha }\)-fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the \(\varvec{\alpha }\)-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 \({\mathcal {C}}[0,1]\).

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We would like to thank Dr. P. Viswanathan for his valuable comments during the preparations of this manuscript. The second author is thankful for the project: MTR/2017/000574 - MATRICS from the Science and Engineering Research Board (SERB), Government of India. We would like to thank anonymous reviewers for several constructive suggestions that greatly contributed to improve the paper.

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Correspondence to Sangita Jha.

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Jha, S., Chand, A.K.B. & Navascués, M.A. Approximation by shape preserving fractal functions with variable scalings. Calcolo 58, 8 (2021).

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  • Fractals
  • Fractal interpolation functions
  • Smoothing
  • Shape preservation
  • Chebyshev system
  • Müntz approximation

Mathematics Subject Classification

  • 28A80
  • 41A15
  • 41A29
  • 41A30
  • 41A50
  • 42A15