Skip to main content
SpringerLink
Log in

Gegenbauer Weight Functions Admitting L 2 Duffin and Schaeffer Type Inequalities

  • Conference paper
Applications and Computation of Orthogonal Polynomials

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 131))

  • 339 Accesses

Abstract

For a given set Ω of n + 1 distinct points in [-1, 1] and algebraic polynomial Q n of degree n, sufficient conditions are found ensuring that for every other polynomial f of degree at most n the inequality |f|≤ |Q n | on Ω induces inequalities between some weighted L 2 norms of the derivatives of f and Q n

Article Footnote

1The second author was supported by a grant of the Royal Society and by the Bulgarian Ministry of Education, Science, and Technologies, under Grant MM-414/94.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Bojanov, An inequality of Duffin and Schaeffer type, East J. Approx., 1 (1995), 37–46.

    MathSciNet  MATH  Google Scholar 

  2. B. Bojanov and G. NikolovDuffin and Schaeffer type inequality for ultraspherical polynomials, J. Approx. Theory, 84 (1996), 129–138.

    Article  MathSciNet  MATH  Google Scholar 

  3. D.K. Dimitrov, Markov inequalities for weight functions of Chebyshev type, J. Approx. Theory, 83 (1995), 175–181.

    Article  MathSciNet  MATH  Google Scholar 

  4. R.J. Duffin and A.C. Schaeffer, A refinement of an inequality of the brothers Markoff, Trans. Amer. Math. Soc., 50 (1941), 517–528.

    Article  MathSciNet  Google Scholar 

  5. A. Guessab and Q.I. RahmanQuadrature formulae and polynomial inequalities, J. Approx. Theory, 90 (1997), 255–282.

    Article  MathSciNet  MATH  Google Scholar 

  6. V.A. MarkovOn functions least deviated from zero in given interval, St. Petersburg, 1892 (in Russian). German translation: Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann., 77 (1916), 213–258.

    Article  MathSciNet  Google Scholar 

  7. G. Nikolov, Weighted L 2 inequalities of Duffin and Schaeffer type via quadrature formulae, submitted for publication.

    Google Scholar 

  8. T.J. Rivlin, The Chebyshev polynomials, Pure and Applied Mathematics, Wiley, New York, 1974.

    MATH  Google Scholar 

  9. A.Yu. Shadrin, Interpolation with Lagrange polynomials. A simple proof of Markov inequality and some of its generalizations, Approx. Theory Appl., 8 (1992), 51–61.

    MathSciNet  MATH  Google Scholar 

  10. G. Szegö, Orthogonal Polynomials,Amer. Math. Soc. Colloquium Publ. 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975.

    Google Scholar 

  11. A.K. Varma, On some extremal properties of algebraic polynomials, J. Approx. Theory, 69 (1992), 48–54.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing, University of Bradford, Bradford, BD7 1DP, England

    David B. Hunter

  2. Department of Mathematics, University of Sofia, Sofia, BG-1126, Bulgaria

    Geno Nikolov

Authors
  1. David B. Hunter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geno Nikolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Sciences, Purdue University, West Lafayette, IN, 47907-1398, USA

    Walter Gautschi

  2. Institute for Applied Mathematics, University of Hamburg, Bundesstr. 55, D-20146, Hamburg, Germany

    Gerhard Opfer

  3. Computer Science Department, Stanford University, Stanford, CA, 94305, USA

    Gene H. Golub

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Basel AG

About this paper

Cite this paper

Hunter, D.B., Nikolov, G. (1999). Gegenbauer Weight Functions Admitting L 2 Duffin and Schaeffer Type Inequalities. In: Gautschi, W., Opfer, G., Golub, G.H. (eds) Applications and Computation of Orthogonal Polynomials. International Series of Numerical Mathematics, vol 131. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8685-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8685-7_8

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9728-0

  • Online ISBN: 978-3-0348-8685-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation