Advertisement

Constructive Approximation

, Volume 31, Issue 2, pp 259–288 | Cite as

Some Extremal Functions in Fourier Analysis, III

  • Emanuel Carneiro
  • Jeffrey D. Vaaler
Article

Abstract

We obtain the best approximation in L 1(ℝ), by entire functions of exponential type, for a class of even functions that includes e λ|x|, where λ>0, log |x| and |x| α , where −1<α<1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.

Keywords

Approximation Entire functions Exponential type 

Mathematics Subject Classification (2000)

41A30 41A52 42A05 41A05 41A44 42A10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barton, J.T., Montgomery, H.L., Vaaler, J.D.: Note on a Diophantine inequality in several variables. Proc. Am. Math. Soc. 129, 337–345 (2000) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Carneiro, E., Vaaler, J.D.: Some extremal functions in Fourier analysis, II. Trans. Am. Math. Soc. (to appear) Google Scholar
  3. 3.
    Ganzburg, M.: The Bernstein constant and polynomial interpolation at the Chebyshev nodes. J. Approx. Theory 119, 193–213 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ganzburg, M., Lubinsky, D.S.: Best approximating entire functions to |x|α in L 2. Contemp. Math. 455, 93–107 (2008) MathSciNetGoogle Scholar
  5. 5.
    Graham, S.W., Vaaler, J.D.: A class of extremal functions for the Fourier transform. Trans. Am. Math. Soc. 265, 283–302 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Holt, J., Vaaler, J.D.: The Beurling-Selberg extremal functions for a ball in the Euclidean space. Duke Math. J. 83, 203–247 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Li, X.J., Vaaler, J.D.: Some trigonometric extremal functions and the Erdös–Turán type inequalities. Indiana Univ. Math. J. 48(1), 183–236 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Littmann, F.: Entire approximations to the truncated powers. Constr. Approx. 22(2), 273–295 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Littmann, F.: Entire majorants via Euler-Maclaurin summation. Trans. Am. Math. Soc. 358(7), 2821–2836 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lubinsky, D.S.: On the Bernstein constants of polynomial approximation. Constr. Approx. 25, 303–366 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Montgomery, H.L.: The analytic principle of the large sieve. Bull. Am. Math. Soc. 84(4), 547–567 (1978) zbMATHCrossRefGoogle Scholar
  12. 12.
    Plancherel, M., Polya, G.: Fonctions entiéres et intégrales de Fourier multiples (Seconde partie). Comment. Math. Helv. 10, 110–163 (1938) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) zbMATHGoogle Scholar
  14. 14.
    Selberg, A.: Lectures on sieves. In: Atle Selberg: Collected Papers, vol. II, pp. 65–247. Springer, Berlin (1991) Google Scholar
  15. 15.
    Vaaler, J.D.: Some extremal functions in Fourier analysis. Bull. Am. Math. Soc. 12, 183–215 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980) zbMATHGoogle Scholar
  17. 17.
    Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (1959) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

Personalised recommendations