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A Fractal Version of a Bivariate Hermite Polynomial Interpolation

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Abstract

This note is concerned with the construction of a class of bivariate fractal functions interpolating a Hermite data set over a rectangular grid. To this end, we apply the general theory of smoothness preserving “fractal perturbation” of a bivariate continuous function to a Hermite interpolation formula in two variables studied by Chawla et al. [J. Aust. Math. Society, 18 (1974), pp. 402-410], which in turn is a slight generalization of Ahlin’s bivariate Hermite interpolation formula that made its debut in [Math. Comp., 18 (1964), pp. 264-273]. An upper bound for the error in approximating a sufficiently smooth function with the proposed bivariate Hermite fractal interpolation function is hinted at. The polynomial, Hermite and lacunary interpolation methods in the univariate setting were constantly reinvigorated using fractal methodology in the last 2 decades. To the writer’s knowledge, this is the first such study for the bivariate Hermite interpolation with the concept of bivariate fractal perturbation as its theoretical framework.

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Acknowledgements

Subsequent to the interim acceptance of this work, the author noticed the article [15] which deal with the same subject matter addressed in the current note. The approach taken in the aforementioned article is quite different from that of ours. This work was partially supported by SERB MATRICS under Project No. MTR/2020/000457.

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

    P. Viswanathan

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Correspondence to P. Viswanathan.

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Viswanathan, P. A Fractal Version of a Bivariate Hermite Polynomial Interpolation. Mediterr. J. Math. 18, 225 (2021). https://doi.org/10.1007/s00009-021-01871-w

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