Skip to main content

Closure properties of sequences of exponentals { exp (i λn t) }

  • Conference paper
  • First Online:
  • 371 Accesses

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 419))

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (USA)
  • 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. Bernstein, S.: Sur l'ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné.-Acad. Roy. Belg. Cl. Sci. Mém. Coll. in-4o (2) 4:1, 1922 [1912].

    Google Scholar 

  2. Boas, R. P., Jr.: Entire functions.-Pure and Applied Mathematics 5. Academic Press Inc., Publishers, New York, 1954.

    Google Scholar 

  3. Carleman, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen.-Ark. Mat. Astronom. Fys. 17:9, 1922/1923.

    MathSciNet  Google Scholar 

  4. Clarkson, J. A., and P. Erdős: Approximation by polynomials.-Duke Math. J. 10, 1943, pp.5–11.

    Article  MathSciNet  MATH  Google Scholar 

  5. Crum, M. M.: On the theorems of Müntz and Szász.-J. London Math. Soc. 31, 1956, pp.433–437.

    Article  MathSciNet  MATH  Google Scholar 

  6. -"-Corrigendum and addendum: On the theorem of Müntz and Szász.-J. London Math. Soc. 32, 1957, p.512.

    Article  MathSciNet  MATH  Google Scholar 

  7. Feller, W.: On Müntz' theorem and completely monotone functions.-Amer. Math. Monthly 75, 1968, pp.342–350.

    Article  MathSciNet  MATH  Google Scholar 

  8. Forst, W.: Ein funktionentheoretischer Beweis des Satzes von Müntz.-Manuscripta Math. 3, 1970, pp.357–374.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ingham, A. E.: Some trigonometrical inequalities with applications to the theory of series.-Math. Z. 41, 1936, pp.367–379.

    Article  MathSciNet  MATH  Google Scholar 

  10. Kahane, J. P.: Sur quelques problèmes d'unicité et de prolongement relatifs aux fonctions approchables par des sommes d'exponentielles.-Ann. Inst. Fourier (Grenoble) 5, 1955, pp.39–130.

    Article  MathSciNet  MATH  Google Scholar 

  11. -"-Lectures on mean periodic functions.-Lectures on Mathematics and Physics, Mathematics 15. Tata Institute of Fundamental Research, Bombay, 1959.

    MATH  Google Scholar 

  12. Korevaar, J.: A characterization of the sub-manifold of C[a,b] spanned by \(\left\{ {x^{n_k } } \right\}\).-Proc. Akad. Wetensch. Proc. Sect. Sci. 50, 1947, pp.750–758.=Nederl. Akad. Wetensch. Indag. Math. 9, 1947, pp.360–368.

    MathSciNet  MATH  Google Scholar 

  13. Левин, Б. Я.: Распределение корней целых функций.-Государственное Нэдательство Технико-Теоретиче ской Литературы, Москва/Ленинград, 1956.

    Google Scholar 

  14. -"-[B. J. Lewin]: Nullstellenverteilung ganzer Funktionen.-Mathematische Lehrbücher und Monographien II:XIV. Akademie-Verlag, Berlin, 1962. [German translation of [12].]

    Google Scholar 

  15. -"-[B. Ja. Levin]: Distribution of zeros of entire functions.-Translations of Mathematical Monographs 6. American Mathematical Society, Providence (N. J.), 1964. [English translation of [12].]

    Google Scholar 

  16. Levinson, N.: Gap and density theorems.-Colloquium Publications XXVI. American Mathematical Society, New York City, 1940.

    Book  MATH  Google Scholar 

  17. Luxemburg, W. A. J., and J. Korevaar: Entire functions and Müntz-Szász type approximation.-Trans. Amer. Math. Soc. 157, 1971, pp.23–37.

    MathSciNet  MATH  Google Scholar 

  18. Mandelbrojt, S.: Fonctions entières et transformées de Fourier. Applications.-Publications of the Mathematical Society of Japan 10. The Mathematical Society of Japan, Tokyo, 1967.

    Google Scholar 

  19. Müntz, Ch. H.: Über den Approximationssatz von Weierstraß.-Mathematische Abhandlungen Hermann Amandus Schwarz zu seinem fünfzigjährigen Doktorjubileum am 6. August 1914 gewidmet von Freunden und Schülern, pp. 303–312. Verlag von Julius Springer, Berlin, 1914.

    Google Scholar 

  20. Paley, R.E.A.C., and N. Wiener: Fourier transforms in the complex domain.-Colloquium Publications XIX. American Mathematical Society, New York, 1934.

    MATH  Google Scholar 

  21. Pólya, G.: Über die Existenz unendlich vieler singulärer Punkte auf der Konvergenzgeraden gewisser Dirichletscher Reihen.-Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin, 1923, pp.45–50.

    Google Scholar 

  22. -"-Untersuchungen über Lücken und Singularitäten von Potenzreihen.-Math. Z. 29, 1929, pp.549–640.

    Article  MathSciNet  MATH  Google Scholar 

  23. Schwartz, L.: Approximation d'une fonction quelonque par des sommes d'exponentielles imaginaires.-Ann. Fac. Sci. Univ. Toulouse (4) 6, 1943, pp.111–176.

    MathSciNet  MATH  Google Scholar 

  24. -"-Étude des somme d'exponentielles réelles.-Publications de l'Institut de Mathématique de l'Université de Clermont-Ferrand V. Actualités Sci. Indust. 959. Hermann & Cie, Éditeurs, Paris, 1943.

    Google Scholar 

  25. -"-Étude des sommes d'exponentielles.-[2ième édition.] Publications de l'Institut de Mathématique de l'Université de Strasbourg V. Actualités Sci. Indust. 959. Herman, Paris, 1959.

    Google Scholar 

  26. Spencer, J. P.: Considerations on a Müntz-type problem on the interval [0,∞].-Mitt. Math. Sem. Gießen 84, 1970.

    Google Scholar 

  27. Szász, O.: Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen.-Math. Ann. 77, 1916, pp.482–496.

    Article  MathSciNet  MATH  Google Scholar 

  28. Beurling, A., and P. Malliavin: On the closure of characters and the zeros of entire functions.-Acta Math. 118, 1967, pp.79–93.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Olli Lehto Ilppo Simo Louhivaara Rolf Nevanlinna

Rights and permissions

Reprints and permissions

Copyright information

© 1974 Springer-Verlag

About this paper

Cite this paper

Luxemburg, W.A.J. (1974). Closure properties of sequences of exponentals { exp (i λn t) }. In: Lehto, O., Louhivaara, I.S., Nevanlinna, R. (eds) Topics in Analysis. Lecture Notes in Mathematics, vol 419. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064735

Download citation

  • DOI: https://doi.org/10.1007/BFb0064735

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06965-2

  • Online ISBN: 978-3-540-37907-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics