Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Experiment Videos

The Markoff-Duffin-Schaeffer inequalities abstracted.

R J Duffin1, L A Karlovitz

  • 1Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1985
PubMed
Summary

The Markoff-Duffin-Schaeffer inequality bounds polynomial derivatives. This study extends these inequalities to generalized polynomials, proving direct analogs hold in an abstract setting.

Related Concept Videos

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Dualizing the Poisson summation formula.

Proceedings of the National Academy of Sciences of the United States of America·1991
Same author

Rubel's universal differential equation.

Proceedings of the National Academy of Sciences of the United States of America·1981
Same author

Bounds for the rth characteristic frequency of a beaded string or of an electrical filter.

Proceedings of the National Academy of Sciences of the United States of America·1980
Same author

Clark's Theorem on linear programs holds for convex programs.

Proceedings of the National Academy of Sciences of the United States of America·1978
Same author

Convex programs having some linear constraints.

Proceedings of the National Academy of Sciences of the United States of America·1977
Same author

Hilbert transforms in yukawan potential theory.

Proceedings of the National Academy of Sciences of the United States of America·1972

Area of Science:

  • Mathematics
  • Approximation Theory
  • Real Analysis

Background:

  • The Markoff-Duffin-Schaeffer inequality bounds the maximum kth derivative of a normalized polynomial on [-1, 1].
  • The bound is given by the maximum of the Chebyshev polynomial of degree n, T(x) = cos(n arccos(x)).
  • Equality holds only for T(x) or -T(x), with normalization at T's extremal points.

Purpose of the Study:

  • To investigate the applicability of Markoff-Duffin-Schaeffer inequalities to generalized polynomials.
  • To demonstrate that direct analogs of these inequalities hold in an abstract setting.
  • To provide a more elementary proof for these generalized inequalities.

Main Methods:

  • Utilizing complex variable theory for the classical inequality proof.

Related Experiment Videos

  • Defining and analyzing generalized polynomials with specific oscillation and approximation properties.
  • Establishing direct analogs of the Markoff-Duffin-Schaeffer inequalities in this abstract framework.

    • Main Results:

    • The direct analogs of the Markoff-Duffin-Schaeffer inequalities are shown to hold for generalized polynomials.
    • The generalized Chebyshev polynomial exhibits characteristic extremal oscillations.
    • The classical inequalities are demonstrated to be a special case of the generalized results.

    Conclusions:

    • The Markoff-Duffin-Schaeffer inequalities have valid extensions to a broader class of functions (generalized polynomials).
    • The proofs for these generalized inequalities are more elementary than the original complex variable approach.
    • This work unifies and extends classical results in approximation theory.