## Abstract and Applied Analysis

### The Multivariate Müntz-Szasz Problem in Weighted Banach Space on ${\mathbb{R}}^{n}$

Xiangdong Yang

#### Abstract

The purpose of this paper is to give an extension of Müntz-Szasz theorems to multivariable weighted Banach space. Denote by $\{{\lambda }_{k}=({\lambda }_{k}^{1},{\lambda }_{k}^{2},...,{\lambda }_{k}^{n}){\}}_{k=1}^{\mathrm{\infty }}$ a sequence of real numbers in ${\mathbb{R}}_{+}^{n}$. The completeness of monomials $\{{t}^{{\lambda }_{k}}\}$ in ${C}_{\alpha }$ is investigated, where ${C}_{\alpha }$ is the weighted Banach spaces which consist of complex continuous functions $f$ defined on ${\mathbb{R}}^{n}$ with $f(t)$ exp$(-\alpha (t))$ vanishing at infinity in the uniform norm.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 342826, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412277003

Digital Object Identifier
doi:10.1155/2014/342826

Mathematical Reviews number (MathSciNet)
MR3214421

Zentralblatt MATH identifier
07022192

#### Citation

Yang, Xiangdong. The Multivariate Müntz-Szasz Problem in Weighted Banach Space on ${\mathbb{R}}^{n}$. Abstr. Appl. Anal. 2014 (2014), Article ID 342826, 7 pages. doi:10.1155/2014/342826. https://projecteuclid.org/euclid.aaa/1412277003

#### References

• S. Hellerstein, “Some analytic varieties in the polydisc and the Müntz-Szasz problem in several variables,” Transactions of the American Mathematical Society, vol. 158, pp. 285–292, 1971.
• S. Ogawa and K. Kitahara, “An extension of Müntz's theorems in multivariables,” Bulletin of the Australian Mathematical Society, vol. 36, pp. 375–387, 1987.
• A. Kroó, “A geometric approach to the multivariate Müntz problem,” Proceedings of the American Mathematical Society, vol. 121, pp. 199–208, 1994.
• G. T. Deng, “Incompleteness and closure of a linear span of exponential system in a weighted Banach space,” Journal of Approximation Theory, vol. 125, no. 1, pp. 1–9, 2003.
• G. Deng, “Incompleteness and minimality of complex exponential system,” Science in China. Series A: Mathematics, vol. 50, no. 10, pp. 1467–1476, 2007.
• X. Yang, “Incompleteness of exponential system in the weighted Banach space,” Journal of Approximation Theory, vol. 153, no. 1, pp. 73–79, 2008.
• X. Yang and J. Tu, “On the completeness of the system $\{{t}^{{\lambda }_{n}}\}$ in ${C}_{0}$(\emphE),” Journal of Mathematical Analysis and Applications, vol. 368, no. 2, pp. 429–437, 2010.
• X. D. Yang, “On the completeness of the system ${t}^{{\lambda }_{n}}$ log$^{j}$ t in ${C}_{0}$(\emphE),” Czechoslovak Mathematical Journal, vol. 62, pp. 361–379, 2012.
• X. D. Yang, “Exponential approximation of weighted Banach space on ${\mathbb{R}}^{n}$,” Glasgow Mathematical Journal, vol. 55, pp. 115–121, 2013.
• P. Koosis, The Logarithmic Integral, Volume I, Cambridge University Press, Cambridge, UK, 1988.
• P. Malliavin, “Sur quelques procédés d'extrapolation,” Acta Mathematica, vol. 93, no. 1, pp. 179–255, 1955.
• G. Deng, “On weighted polynomial approximation with gaps,” Nagoya Mathematical Journal, vol. 178, pp. 55–61, 2005.
• G. T. Deng, “Weighted exponential polynomial approximation,” Science in China, vol. 46, pp. 280–287, 2003.
• N. V. Govorov, Riemann's Boundary Problem with Infinite Index, Birkhäuser, Berlin, Germany, 1994.
• W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^{n}$, Springer, New York, NY, USA, 1980.
• M. de Jeu, “Determinate multidimensional measures, the extended carleman theorem and quasi-analytic weights,” The Annals of Probability, vol. 31, no. 3, pp. 1205–1227, 2003.
• M. de Jeu, “Subspaces with equal closure,” Constructive Approximation, vol. 20, no. 1, pp. 93–157, 2004.
• W. Rudin, Real and Complex Analysis, China Machine Press, Beijing, China, 2003.
• P. Duren, Theory of H$^{p}$ Spaces, Academic Press, New York, NY, USA, 1970. \endinput