Skip to main content
Springer Nature Link
Log in

On the number of roots of exp-trig polynomials

Über die Anzahl der Nullstellen von polynomial-trigonometrischen Exponentialsummen

  • Short Communication
  • Published:
Computing Aims and scope Submit manuscript

Abstract

The number of roots, on an interval of lengthT, ofc′ e At b≢0 is at most (n-1)[T ω/π]*, wheren is the dimension ofA, b, c; ω is the largest imaginary part of eigenvalue ofA, and [x]* denotes the smallest integerk≥x.

Zusammenfassung

Eine Funktion der Formc′ e At b≢0 hat höchstens (n-1)[T ω/π]* Nullstellen in einem Intervall der LängeT, wobein die Dimension vonA, b, c ist, ω der größte Imaginärteil der Eigenwerte vonA und [x]* die kleinste ganze Zahlk≥x.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Chukwu, E. N., Hájek, O.: Disconjugacy and optimal control (to appear in Journ. Opt. Th. and Appl.).

  2. Collatz, L., Krabs, W.: Approximationstheorie. Stuttgart: Teubner 1973.

    Google Scholar 

  3. Hájek, O.: Optimal controls with side constraints (in preparation).

  4. Hájek, O.: Terminal manifolds and switching locus. Math. Syst. Theory6, 289–301 (1973).

    Google Scholar 

  5. Kolumbán, I.: Über die nichtlineare trigonometrische Approximation, pp. 69–72, in: Numerische Methoden der Approximationstheorie, Bd. 2 (Collatz, L., Meinardus, G., eds.). Basel: Birkhäuser 1975.

    Google Scholar 

  6. Lee, E. B., Markus, L.: Foundations of Optimal Control Theory. New York: Wiley 1967.

    Google Scholar 

  7. Levinson, N.: Minimax Liapunov and “bang-bang”. J. Diff. Equations2, 218–241 (1966).

    Google Scholar 

  8. Meinardus, G.: Über ein Problem von L. Collatz. Computing8 250–254 (1971).

    Google Scholar 

  9. Pólya, G., Szegö, G.: Problems and Theorems in Analysis, Vol. 1. Berlin-Heidelberg-New York: Springer 1972.

    Google Scholar 

  10. Yeung, D. S.: Synthesis of time-optimal control. Ph. D. Thesis, Case Western Reserve University, Cleveland, Ohio, 1974.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Case Western Reserve University, 44106, Cleveland, OH, USA

    O. Hájek

Authors
  1. O. Hájek
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper was prepared while the author held a Humboldt Award at TH Darmstadt, Fachbereich Mathematik, Arbeitsgruppe 10.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hájek, O. On the number of roots of exp-trig polynomials. Computing 18, 177–183 (1977). https://doi.org/10.1007/BF02243627

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02243627

Keywords