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Shape preservation of scientific data through rational fractal splines

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Abstract

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of \(\mathcal C ^1\)- rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form \(\frac{p_i(x)}{q_i(x)},\) where \(p_i(x)\) and \(q_i(x)\) are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in \(\mathcal C ^2\) is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.

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Acknowledgments

The partial support of the Department of Science and Technology of Govt. of India (SERC DST Project No. SR/S4/MS: 694/10) is gratefully acknowledged. The authors are thankful to the referees for their valuable suggestions to improve the presentation of the article.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    A. K. B. Chand & N. Vijender

  2. Departmento de Matemática Aplicada, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, C/ María de Luna 3, Zaragoza,  50018, Spain

    M. A. Navascués

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  1. A. K. B. Chand
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  2. N. Vijender
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  3. M. A. Navascués
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Correspondence to A. K. B. Chand.

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Chand, A.K.B., Vijender, N. & Navascués, M.A. Shape preservation of scientific data through rational fractal splines. Calcolo 51, 329–362 (2014). https://doi.org/10.1007/s10092-013-0088-2

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  • DOI: https://doi.org/10.1007/s10092-013-0088-2

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