## Multivariate polynomial approximation in the hypercube

HTML articles powered by AMS MathViewer

- by Lloyd N. Trefethen PDF
- Proc. Amer. Math. Soc.
**145**(2017), 4837-4844 Request permission

## 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 $x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.

## References

- V. I. Arnolâ€™d,
*Huygens & Barrow, Newton & Hooke: Pioneers in Mathematical Analysis and Catastrophe Theory from Evolvents to Quasicrystals,*BirkhĂ¤user, 1990. - Serge Bernstein,
*Sur la meilleure approximation de $|x|$ par des polynomes de degrĂ©s donnĂ©s*, Acta Math.**37**(1914), no.Â 1, 1â€“57 (French). MR**1555093**, DOI 10.1007/BF02401828 - Salomon Bochner and William Ted Martin,
*Several Complex Variables*, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR**0027863** - John P. Boyd,
*Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms*, J. Engrg. Math.**63**(2009), no.Â 2-4, 355â€“399. MR**2486453**, DOI 10.1007/s10665-008-9241-3 - T. A. Driscoll, N. Hale, and L. N. Trefethen,
*Chebfun Userâ€™s Guide,*Pafnuty Publications, Oxford, 2014. See also www.chebfun.org. - Nicholas Hale and Lloyd N. Trefethen,
*New quadrature formulas from conformal maps*, SIAM J. Numer. Anal.**46**(2008), no.Â 2, 930â€“948. MR**2383217**, DOI 10.1137/07068607X - Arnold R. Krommer and Christoph W. Ueberhuber,
*Computational integration*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. MR**1625683**, DOI 10.1137/1.9781611971460 - J. C. Mason,
*Near-best multivariate approximation by Fourier series, Chebyshev series and Chebyshev interpolation*, J. Approx. Theory**28**(1980), no.Â 4, 349â€“358. MR**589990**, DOI 10.1016/0021-9045(80)90069-6 - J. C. Maxwell,
*On approximate multiple integration between limits of summation*, Proc. Camb. Phil. Soc. 3 (1877), 39â€“47. - A. F. Timan,
*Theory of approximation of functions of a real variable*, Dover Publications, Inc., New York, 1994. Translated from the Russian by J. Berry; Translation edited and with a preface by J. Cossar; Reprint of the 1963 English translation. MR**1262128** - Lloyd N. Trefethen,
*Approximation theory and approximation practice*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR**3012510** - L. N. Trefethen,
*Cubature, approximation, and isotropy in the hypercube*, SIAM Rev., to appear.

## Additional Information

**Lloyd N. Trefethen**- Affiliation: Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
- MR Author ID: 174135
- Email: trefethen@maths.ox.ac.uk
- 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 Unionâ€™s Seventh Framework Programme (FP7/2007â€“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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 4837-4844 - MSC (2010): Primary 41A63
- DOI: https://doi.org/10.1090/proc/13623
- MathSciNet review: 3691999