On the density of multivariate polynomials with varying weights
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- by András Kroó and József Szabados
- Proc. Amer. Math. Soc. 151 (2023), 1921-1935
- DOI: https://doi.org/10.1090/proc/14882
- Published electronically: February 2, 2023
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Abstract:
In this paper we consider multivariate approximation by weighted polynomials of the form $w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x})$, where $p_n$ is a multivariate polynomial of degree at most $n$, $w$ is a given nonnegative weight with nonempty zero set, and $\gamma _n\uparrow \infty$. We study the question if every continuous function vanishing on the zero set of $w$ is a uniform limit of weighted polynomials $w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x})$. It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that $\gamma _n=o(n).$References
- M. von Golitschek, Approximation by incomplete polynomials, J. Approx. Theory 28 (1980), no. 2, 155–160. MR 573329, DOI 10.1016/0021-9045(80)90086-6
- Geza Freud, Markov-Bernstein type inequalities in $L_{p}(-\infty ,\infty )$, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York-London, 1976, pp. 369–377. MR 430187
- Géza Freud, On Markov-Bernstein-type inequalities and their applications, J. Approximation Theory 19 (1977), no. 1, 22–37. MR 425426, DOI 10.1016/0021-9045(77)90026-0
- Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR 1261635, DOI 10.1007/978-3-662-02888-9
- András Kroó, Weierstrass type approximation by weighted polynomials in $\Bbb R^d$, J. Approx. Theory 246 (2019), 85–101. MR 3983042, DOI 10.1016/j.jat.2019.07.005
- András Kroó and Vilmos Totik, Density of bivariate homogeneous polynomials on non-convex curves, Proc. Amer. Math. Soc. 147 (2019), no. 1, 167–177. MR 3876740, DOI 10.1090/proc/14237
- A. L. Levin and D. S. Lubinsky, Canonical products and the weights $\textrm {exp}(-|x|^\alpha ),\;\alpha >1$, with applications, J. Approx. Theory 49 (1987), no. 2, 149–169. MR 874951, DOI 10.1016/0021-9045(87)90085-2
- G. G. Lorentz, Approximation of functions, 2nd ed., Chelsea Publishing Co., New York, 1986. MR 917270
- G. G. Lorentz, Approximation by incomplete polynomials (problems and results), Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York-London, 1977, pp. 289–302. MR 467089
- G. G. Lorentz, Problems for incomplete polynomials, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980) Academic Press, New York-London, 1980, pp. 41–73. MR 602705
- D. S. Lubinsky, A survey of weighted polynomial approximation with exponential weights, Surv. Approx. Theory 3 (2007), 1–105. MR 2276420
- Doron S. Lubinsky and Vilmos Totik, Weighted polynomial approximation with Freud weights, Constr. Approx. 10 (1994), no. 3, 301–315. MR 1291052, DOI 10.1007/BF01212563
- D. S. Lubinsky, Jackson and Bernstein theorems for the weight $\exp (-|x|)$ on $\Bbb R$, Israel J. Math. 153 (2006), 193–219. MR 2254642, DOI 10.1007/BF02771783
- 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, DOI 10.1007/978-3-662-03329-6
- E. B. Saff and R. S. Varga, Uniform approximation by incomplete polynomials, Internat. J. Math. Math. Sci. 1 (1978), no. 4, 407–420. MR 517944, DOI 10.1155/S0161171278000411
- Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789, DOI 10.1007/BFb0076133
Bibliographic Information
- András Kroó
- Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1053 Hungary; and Budapest University of Technology and Economics, Department of Analysis, Budapest, 1111 Hungary
- Email: kroo.andras@renyi.mta.hu
- József Szabados
- Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1053 Hungary
- Email: szabados.jozsef@renyi.mta.hu
- Received by editor(s): March 21, 2019
- Received by editor(s) in revised form: September 19, 2019
- Published electronically: February 2, 2023
- Additional Notes: The research of both authors was supported by NKFI Grant No. K128922.
- Communicated by: Yuan Xu
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1921-1935
- MSC (2010): Primary 41A10, 41A63
- DOI: https://doi.org/10.1090/proc/14882
- MathSciNet review: 4556189