Advertisement

On the completeness of algebraic polynomials in the spaces L p (ℝ, dμ)

  • A. G. Bakan
Article
  • 25 Downloads

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.

Keywords

Entire Function Moment Problem Algebraic Polynomial Discrete Measure Entire Transcendental Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    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).zbMATHMathSciNetGoogle Scholar
  2. 2.
    L. de Branges, “The Bernstein problem,” Proc. Amer. Math. Soc., 10, 825–832 (1959).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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).zbMATHGoogle Scholar
  4. 4.
    H. Pollard, “Solution of Bernstein’s approximation problem,” Proc. Amer. Math. Soc., 4, 869–875 (1959).CrossRefMathSciNetGoogle Scholar
  5. 5.
    S. N. Mergelyan, “Weighted polynomial approximations,” Usp. Mat. Nauk, 11, 107–152 (1956).MathSciNetGoogle Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    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).CrossRefMathSciNetGoogle Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    G. P. Akilov and L. V. Kantorovich, Functional Analysis in Normed Spaces, Macmillan, New York (1964).zbMATHGoogle Scholar
  10. 10.
    M. Riesz, “Sur le problème des moments et le théorème de Parseval correspondant,” Acta Szeged Sect. Math., 1, 209–225 (1923).Google Scholar
  11. 11.
    C. Berg and M. Thill, “Rotation invariant moment problem,” Acta Math., 167, 207–227 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. W. Widder, The Laplace Transform, Vol. 1, Princeton University, Princeton (1941).zbMATHGoogle Scholar
  13. 13.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1953).Google Scholar
  14. 14.
    M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions, National Bureau of Standards, U.S. Department Commerce (1964).Google Scholar
  15. 15.
    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).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    C. Berg and J. P. R. Christensen, “Exposants critiques dans le problème des moments,” C. R. Acad. Sci. Paris, 296, 661–663 (1983).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • A. G. Bakan
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

Personalised recommendations