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In reference to a self-referential approach towards smooth multivariate approximation

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Abstract

Approximation of a multivariate function is an important theme in the field of numerical analysis and its applications, which continues to receive a constant attention. In this paper, we provide a parameterized family of self-referential (fractal) approximants for a given multivariate smooth function defined on an axis-aligned hyper-rectangle. Each element of this class preserves the smoothness of the original function and interpolates the original function at a prefixed gridded data set. As an application of this construction, we deduce a fractal methodology to approach a multivariate Hermite interpolation problem. This part of our paper extends the classical bivariate Hermite’s interpolation formula by Ahlin (Math. Comp. 18:264–273, 1964) in a twofold sense: (i) records, in particular, a multivariate generalization of this bivariate interpolation theory; (ii) replaces the unicity of the Hermite interpolant with a parameterized family of self-referential Hermite interpolants which contains the multivariate analogue of Ahlin’s interpolant as a particular case. Some related aspects including the approximation by multivariate self-referential functions preserving Popoviciu convexity are given, too.

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Acknowledgements

The second author is thankful to the project CRG/2020/002309 from the Science and Engineering Research Board (SERB), Government of India.

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  1. Department of Mathematics, IIT Delhi, New Delhi, 110016, India

    K. K. Pandey & P. Viswanathan

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  1. K. K. Pandey
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  2. P. Viswanathan
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Correspondence to P. Viswanathan.

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Pandey, K.K., Viswanathan, P. In reference to a self-referential approach towards smooth multivariate approximation. Numer Algor 91, 251–281 (2022). https://doi.org/10.1007/s11075-022-01261-7

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