Skip to main content

Bernstein and markov inequalities for constrained polynomials

Approximation And Interpolation Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1105)

Abstract

Pointwise and uniform bounds are determined for the derivatives of real algebraic polynomials p(x) which on the interval [−1,1] satisfy (1−x2)λ/2|p(x)| ≤ 1, λ a fixed positive integer. The pointwise bounds are investigated with regard to their sharpness while the uniform bounds are shown to be best possible in an asymptotic sense.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Bernstein, Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, Mem. Acad. Roy. Belg. (2) 4 (1912), 1–103.

    MATH  Google Scholar 

  2. J.H.B. Kemperman and G.G. Lorentz, Bounds for polynomials with applications, Neder. Akad. Wetensch. Proc. Ser. A. 82 (1979), 13–26.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. M.A. Lachance, E.B. Saff, and R.S. Varga, Bounds for incomplete polynomials vanishing at both endpoints of an interval, in "Constructive Approaches to Mathematical Models" (C.V. Coffman and G.J. Fix, Eds.), 421–437, Academic Press, New York, 1979.

    Google Scholar 

  4. M.A. Lachance, E.B. Saff, and R.S. Varga, Inequalities for polynomials with a prescribed zero, Math. Zeit. 168 (1979), 105–116.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. G.G. Lorentz, Approximation by incomplete polynomials (problems and results), in "Padé and Rational Approximation: Theory and Applications" (E.B. Saff and R.S. Varga, Eds.), 289–302, Academic Press, New York, 1977.

    CrossRef  Google Scholar 

  6. A. Markov, On a certain problem of D.I. Mendeleieff, Utcheniya Zapiski Imperatorskoi Akademii Nauk (Russia), 62 (1889), 1–24.

    Google Scholar 

  7. D.J. Newman and T.J. Rivlin, On polynomials with curved majorants, Can. J. Math., 34 (1982) 961–968.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. R. Pierre and Q.I. Rahman, On a problem of Turán about polynomials, Proc. Amer. Math. Soc., 56 (1976), 231–238.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. R. Pierre and Q.I. Rahman, On a problem of Turán about polynomials, II, Can. J. Math., 33 (1981), 701–733.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Q.I. Rahman, On a problem of Turán about polynomials with curved majorants, Trans. Amer. Math. Soc., 163 (1972), 447–518.

    MathSciNet  MATH  Google Scholar 

  11. I. Schur, Über das maximum des absoluten betrages eines polynoms in einem gegebenen interval, Math. Zeit., 4 (1919), 271–287.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. G. Szegö, Orthogonal Polynomials, Colloquium Publication, Vol. 23, 4th ed., Providence, Rhode Island, American Mathematical Society, 1978.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Lachance, M.A. (1984). Bernstein and markov inequalities for constrained polynomials. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072405

Download citation

  • DOI: https://doi.org/10.1007/BFb0072405

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13899-0

  • Online ISBN: 978-3-540-39113-5

  • eBook Packages: Springer Book Archive