Mathematische Annalen

, Volume 230, Issue 1, pp 57–74 | Cite as

Trigonometric approximation with exponential error orders

I. construction of asymptotically optimal processes; generalized de la Vallée Poussin sums
  • Wolfgang Dahmen


Error Order Trigonometric Approximation Exponential Error 
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  1. 1.
    Achieser, N. I.: Vorlesungen über Approximationstheorie. Berlin: Akademie Verlag 1967Google Scholar
  2. 2.
    Berens, H.: Interpolationsmethoden zur Behandlung von Approximationsprozessen auf Banachräumen. Lecture Notes in Mathematics 64. Berlin, Heidelberg, New York: Springer 1967Google Scholar
  3. 3.
    Butzer, P. L., Nessel, R. J.: Fourier analysis and approximation. Vol. I. Basel: Birkhäuser 1971Google Scholar
  4. 4.
    Dahmen, W.: Trigonometric approximation with exponential error orders II: Properties of asymptotically optimal processes; impossibility of arbitrarily good error estimates. (to appear)Google Scholar
  5. 5.
    Dahmen, W.: Trigonometric approximation with exponential error orders III: Criteria for asymptotically optimal processes. (to appear)Google Scholar
  6. 6.
    Dahmen, W., Görlich, E.: A conjecture of Golomb on optimal and nearly-optimal approximation. Bull. Amer. Math. Soc.80, 1199–1202 (1974)Google Scholar
  7. 7.
    Dahmen, W., Görlich, E.: Asymptotically optimal linear approximation processes and a conjecture of Golomb. In: Linear Operators and Approximation II (ed. by P. L. Butzer and B. Sz.-Nagy) ISNM Vol. 25. Basel: Birkhäuser 1974Google Scholar
  8. 8.
    Dahmen, W., Görlich, E.: Best approximation with exponential orders and intermediate spaces. Math. Z.148, 7–21 (1976)Google Scholar
  9. 9.
    Dahmen, W., Görlich, E.: The characterization problem for best approximation with exponential error orders and evaluation of entropy. Math. Nachr. 1976 (in print)Google Scholar
  10. 10.
    du Bois-Reymond, P.: Sur la grandeur relative des infinis des fonctions. Annali di Mat. Pura e Appl.4, 338–353 (1971)Google Scholar
  11. 11.
    Görlich, E.: Logarithmic and exponential variants of Bernstein's inequality and generalized derivatives. In: Linear operators and approximation (ed. by P. L. Butzer, J.-P. Kahane and B. Sz.-Nagy), ISNM 20. Basel: Birkhäuser 1972Google Scholar
  12. 12.
    Golomb, M.: Optimal and nearly-optimal linear approximation. In: Approximation of functions (ed. by H. L. Garabedian), Amsterdam: Elsevier 1965Google Scholar
  13. 13.
    Golomb, M.: Lectures on theory of approximation. Mimeographed Lecture Notes. Argonne National Laboratory 1962Google Scholar
  14. 14.
    Korneičuk, N. P.: On methods of investigating extremal problems in the theory of best approximation. Russian Math. Surveys29, 7–43 (1974)Google Scholar
  15. 15.
    Leindler, L.: On summability of Fourier series. Acta. Sci. Math. (Szeged)29, 147–162 (1968)Google Scholar
  16. 16.
    Lorentz, G. G.: Metric entropy, widths and superpositions of functions. Amer. Math. mthly.69, 469–485 (1962)Google Scholar
  17. 17.
    Lorentz, G. G.: Metric entropy and approximation. Bull. Amer. Math. Soc.72, 903–937 (1966)Google Scholar
  18. 18.
    Stečkin, S. B.: On de la Vallée Poussin sums. Dokl. Akad. Nauk SSSR (NS)80, 545–548 (1951)Google Scholar
  19. 19.
    Teljakovskiñ, S. A.: Approximation of differentiable functions by de la Vallée Poussin's sums. Dokl. Akad. Nauk SSSR121, 426–429 (1958)Google Scholar
  20. 19.
    Timan, A. F.: Theory of approximation of functions of a real variable. Oxford: Pergamon Press 1963Google Scholar
  21. 20.
    Zygmund, A.: Trigonometric series I. Cambridge: Cambridge University Press 1959Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Wolfgang Dahmen
    • 1
  1. 1.Institut für Angewandte Mathematik der Universität, BonnBonnFederal Republic of Germany

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