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Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

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Abstract

Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

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Notes

  1. in the proof we show that the distance to the nearest integer is less than \(\frac{1}{2}\) so this is well defined.

  2. To simplify notation, we will not introduce new symbols to distinguish between an element of \(\mathbb {K}[x^{\pm }]\) and its image in \(\mathbb {K}[x^{\pm }]/J\).

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Acknowledgements

The authors wish to thank the Fields institute and the organizers of the thematic program on computer algebra where this research was initiated. They also wish to thank Andrew Arnold for discussions on sparse interpolation and the timely pointer on the use of the hypercross in the multivariate case. Thanks also go to the referees of this paper for very useful comments and suggestions.

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  1. INRIA Méditerranée, 06902, Sophia Antipolis, France

    Evelyne Hubert

  2. Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC, 27695-8205, USA

    Michael F. Singer

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  1. Evelyne Hubert
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Communicated by Hans Munthe-Kaas.

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Hubert, E., Singer, M.F. Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials. Found Comput Math 22, 1801–1862 (2022). https://doi.org/10.1007/s10208-021-09535-7

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