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Multivariate polynomial approximation in the hypercube

Author: Lloyd N. Trefethen
Journal: Proc. Amer. Math. Soc. 145 (2017), 4837-4844
MSC (2010): Primary 41A63
Published electronically: June 8, 2017
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Abstract: A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $ s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial, but by the Euclidean degree, defined in terms of the 2-norm rather than the 1-norm of the exponent vector $ \mathbf {k}$ of a monomial $ x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.

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Additional Information

Lloyd N. Trefethen
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom

Received by editor(s): August 7, 2016
Received by editor(s) in revised form: December 6, 2016, and December 12, 2016
Published electronically: June 8, 2017
Additional Notes: The author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 291068
The views expressed in this article are not those of the ERC or the European Commission, and the European Union is not liable for any use that may be made of the information contained here.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2017 American Mathematical Society

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