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A Weierstrass-type theorem for homogeneous polynomials


Authors: David Benko and András Kroó
Journal: Trans. Amer. Math. Soc. 361 (2009), 1645-1665
MSC (2000): Primary 41A10, 31A05; Secondary 52A10, 52A20
DOI: https://doi.org/10.1090/S0002-9947-08-04625-4
Published electronically: October 22, 2008
MathSciNet review: 2457412
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Abstract: By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in $ \mathbb{R}^d$. In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear, this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in $ \mathbb{R}^d$ can be approximated by sums of 2 homogeneous polynomials. This conjecture has not yet been resolved, but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) $ d=2$; 2) convex surfaces in $ \mathbb{R}^d$ with $ C^{1+\epsilon}$ boundary.


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Additional Information

David Benko
Affiliation: Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101
Address at time of publication: Department of Mathematics and Statistics, ILB 325, University of South Alabama, Mobile, Alabama 36688
Email: dbenko@jaguar1.usouthal.edu

András Kroó
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13-15, Hungary
Email: kroo@renyi.hu

DOI: https://doi.org/10.1090/S0002-9947-08-04625-4
Keywords: Weierstrass, uniform approximation, homogeneous polynomials, convex body
Received by editor(s): October 14, 2005
Received by editor(s) in revised form: April 24, 2007
Published electronically: October 22, 2008
Additional Notes: The second author was supported by the OTKA grant # T049196. This research was partially written during this author’s stay at the Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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