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 | References | Similar Articles | Additional Information

Abstract: By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in . 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 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) ; 2) convex surfaces in with boundary.

**1.**David Benko,*Approximation by weighted polynomials*, J. Approx. Theory**120**(2003), no. 1, 153–182. MR**1954937**, https://doi.org/10.1016/S0021-9045(02)00017-5**2.**David Benko,*The support of the equilibrium measure*, Acta Sci. Math. (Szeged)**70**(2004), no. 1-2, 35–55. MR**2071963****3.**D. Benko, S. B. Damelin, and P. D. Dragnev,*On the support of the equilibrium measure for arcs of the unit circle and for real intervals*, Electron. Trans. Numer. Anal.**25**(2006), 27–40. MR**2280361****4.**A. Kroó and J. Szabados,*On density of homogeneous polynomials on convex and star-like surfaces in ℝ^{𝕕}*, East J. Approx.**11**(2005), no. 4, 381–404. MR**2189221****5.**A. B. J. Kuijlaars,*A note on weighted polynomial approximation with varying weights*, J. Approx. Theory**87**(1996), no. 1, 112–115. MR**1410614**, https://doi.org/10.1006/jath.1996.0094**6.**N. I. Muskhelishvili,*Singular integral equations*, Dover Publications, Inc., New York, 1992. Boundary problems of function theory and their application to mathematical physics; Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok; Corrected reprint of the 1953 English translation. MR**1215485****7.**E. B. Saff,*Incomplete and orthogonal polynomials*, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 219–256. MR**754347****8.**Edward B. Saff and Vilmos Totik,*Logarithmic potentials with external fields*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR**1485778****9.**Plamen Simeonov,*A weighted energy problem for a class of admissible weights*, Houston J. Math.**31**(2005), no. 4, 1245–1260. MR**2175434****10.**A. F. Timan,*Theory of Functions of a Real Variable*, (Moscow, 1960) (in Russian).**11.**Vilmos Totik,*Weighted polynomial approximation for convex external fields*, Constr. Approx.**16**(2000), no. 2, 261–281. MR**1735243**, https://doi.org/10.1007/s003659910011**12.**Vilmos Totik,*Weighted approximation with varying weight*, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR**1290789****13.**Péter P. Varjú,*Approximation by homogeneous polynomials*, Constr. Approx.**26**(2007), no. 3, 317–337. MR**2335686**, https://doi.org/10.1007/s00365-006-0639-2

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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.