We prove that the theorem on the incompleteness of polynomials in the space C 0 w established by de Branges in 1959 is not true for the space L p (ℝ, dμ) if the support of the measure μ is sufficiently dense.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Bernstein, “Le problème de l’approximation des fonctions continues sur tout l’axe réel at l’une de ses applications,” Bull. Soc. Math. France, 52, 399–410 (1924).
L. de Branges, “The Bernstein problem,” Proc. Amer. Math. Soc., 10, 825–832 (1959).
N. Akhiezer and S. Bernstein, “Generalization of a theorem on weight functions and application to the moment problem,” Dokl. Akad. Nauk SSSR, 92, 1109–1112 (1953).
H. Pollard, “Solution of Bernstein’s approximation problem,” Proc. Amer. Math. Soc., 4, 869–875 (1959).
S. N. Mergelyan, “Weighted polynomial approximations,” Usp. Mat. Nauk, 11, 107–152 (1956).
M. Sodin and P. Yuditskii, “Another approach to de Branges’ theorem on weighted polynomial approximation,” in: Proceedings of Ashkelon Workshop on Complex Function Theory (Isr. Math. Conf. Proc., May 1996), Vol. 11, American Mathematical Society, Providence (1997), pp. 221–227.
A. Borichev and M. Sodin, “The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line,” J. Anal. Math., 71, 219–264 (1998).
A. G. Bakan, “Polynomial density in L p (R 1, dμ) and representation of all measures which generate a determinate Hamburger moment problem,” in: Approximation, Optimization, and Mathematical Economics, Physica, Heidelberg (2001), pp. 37–46.
G. P. Akilov and L. V. Kantorovich, Functional Analysis in Normed Spaces, Macmillan, New York (1964).
M. Riesz, “Sur le problème des moments et le théorème de Parseval correspondant,” Acta Szeged Sect. Math., 1, 209–225 (1923).
C. Berg and M. Thill, “Rotation invariant moment problem,” Acta Math., 167, 207–227 (1991).
D. W. Widder, The Laplace Transform, Vol. 1, Princeton University, Princeton (1941).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1953).
M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions, National Bureau of Standards, U.S. Department Commerce (1964).
A. Bakan and S. Ruscheweyh, “Representation of measures with simultaneous polynomial denseness in L p (ℝ, dμ), 1 ≤ p < ∞,” Ark. Mat., 43, No. 2, 221–249 (2005).
C. Berg and J. P. R. Christensen, “Exposants critiques dans le problème des moments,” C. R. Acad. Sci. Paris, 296, 661–663 (1983).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 3, pp. 291–301, March, 2009.
Rights and permissions
About this article
Cite this article
Bakan, A.G. On the completeness of algebraic polynomials in the spaces L p (ℝ, dμ). Ukr Math J 61, 347–360 (2009). https://doi.org/10.1007/s11253-009-0221-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0221-1