A Weierstrasstype theorem for homogeneous polynomials
Authors:
David Benko and András Kroó
Journal:
Trans. Amer. Math. Soc. 361 (2009), 16451665
MSC (2000):
Primary 41A10, 31A05; Secondary 52A10, 52A20
Published electronically:
October 22, 2008
MathSciNet review:
2457412
Fulltext PDF Free Access
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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 . 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 starlike 0symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a starlike surface is a convex surface. It has been conjectured by the second author that functions continuous on 0symmetric 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
(2004d:41050), http://dx.doi.org/10.1016/S00219045(02)000175
 2.
David
Benko, The support of the equilibrium measure, Acta Sci. Math.
(Szeged) 70 (2004), no. 12, 35–55. MR 2071963
(2005c:31002)
 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
(2008b:31003)
 4.
A.
Kroó and J.
Szabados, On density of homogeneous polynomials on convex and
starlike surfaces in ℝ^{𝕕}, East J. Approx.
11 (2005), no. 4, 381–404. MR 2189221
(2006i:41027)
 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
(97k:41010), http://dx.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
(94a:45001)
 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
(86b:41029)
 8.
Edward
B. Saff and Vilmos
Totik, Logarithmic potentials with external fields,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 316, SpringerVerlag, Berlin, 1997.
Appendix B by Thomas Bloom. MR 1485778
(99h:31001)
 9.
Plamen
Simeonov, A weighted energy problem for a class of admissible
weights, Houston J. Math. 31 (2005), no. 4,
1245–1260. MR 2175434
(2006i:31002)
 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
(2000j:41010), http://dx.doi.org/10.1007/s003659910011
 12.
Vilmos
Totik, Weighted approximation with varying weight, Lecture
Notes in Mathematics, vol. 1569, SpringerVerlag, Berlin, 1994. MR 1290789
(96f:41002)
 13.
Péter
P. Varjú, Approximation by homogeneous polynomials,
Constr. Approx. 26 (2007), no. 3, 317–337. MR 2335686
(2008i:41011), http://dx.doi.org/10.1007/s0036500606392
 1.
 D. Benko, Approximation by weighted polynomials, J. Approx. Theory 120 (2003), no. 1, 153182. MR 1954937 (2004d:41050)
 2.
 D. Benko, The support of the equilibrium measure, Acta Sci. Math. (Szeged) 70 (2004), no. 12, 3555. MR 2071963 (2005c:31002)
 3.
 D. Benko, S.B. Damelin, 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), 2740. MR 2280361 (2008b:31003)
 4.
 A. Kroó, J. Szabados, On density of homogeneous polynomials on convex and starlike surfaces in , East J. Approx 11 (2005), 381404. MR 2189221 (2006i:41027)
 5.
 A.B.J. Kuijlaars, A note on weighted polynomial approximation with varying weights, J. Approx. Theory 87 (1) (1996), 112115. MR 1410614 (97k:41010)
 6.
 N.I. Muskhelishvili, Singular Integral Equations. Dover, New York, 1992. MR 1215485 (94a:45001)
 7.
 E.B. Saff, Incomplete and orthogonal polynomials. In C.K. Chui, L.L Schumaker, and J.D. Ward, editors, Approximation Theory IV, 219255, Academic Press, New York, 1983. MR 754347 (86b:41029)
 8.
 E.B. Saff, V. Totik, Logarithmic Potentials with External Fields, SpringerVerlag, Berlin, 1997. MR 1485778 (99h:31001)
 9.
 P. Simeonov, A minimal weighted energy problem for a class of admissible weights, Houston J. of Math. 31 (2005), no. 4, 12451260. MR 2175434 (2006i:31002)
 10.
 A. F. Timan, Theory of Functions of a Real Variable, (Moscow, 1960) (in Russian).
 11.
 V. Totik, Weighted polynomial approximation for convex external fields, Constr. Approx. 16 (2) (2000) 261281. MR 1735243 (2000j:41010)
 12.
 V. Totik, Weighted approximation with varying weight. Lecture Notes in Mathematics, 1569, SpringerVerlag, Berlin, 1994. MR 1290789 (96f:41002)
 13.
 P. Varjú, Approximation by homogeneous polynomials, Constr. Approx. 26 (2007), no. 3, 317337. MR 2335686
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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, H1053 Budapest, Reáltanoda u. 1315, Hungary
Email:
kroo@renyi.hu
DOI:
http://dx.doi.org/10.1090/S0002994708046254
PII:
S 00029947(08)046254
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.
