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A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

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Abstract

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a -cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the L p -norm, 1≤p≤∞) for the interpolation error of the -cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys global smoothness. Consequently, our method offers an alternative to the standard moment construction of -cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting -cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results.

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Acknowledgements

The authors are grateful to the anonymous referees for extensive comments and constructive criticisms that improved the presentation of the paper. Special thanks to the referee for bringing the area of recursive subdivision to our attention and suggesting some references. It is our pleasure to thank Professor Rosemary Renaut for the unusually careful editing of the manuscript.

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Authors and Affiliations

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

    A. K. B. Chand & P. Viswanathan

Authors
  1. A. K. B. Chand
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  2. P. Viswanathan
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Correspondence to A. K. B. Chand.

Additional information

Communicated by Rose Marie Renaut.

The first author is thankful to the Department of Science & Technology, India for the SERC DST Project No. SR/S4/MS: 694/10. The research of the second author is partially supported by the Council of Scientific & Industrial Research, India, Grant No. 09/084(0531)/2010-EMR-I. A part of the results in this article was presented at the 10th International Conference of Numerical Analysis and Applied Mathematics, Greece [11].

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Chand, A.K.B., Viswanathan, P. A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects. Bit Numer Math 53, 841–865 (2013). https://doi.org/10.1007/s10543-013-0442-4

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