Advertisement

Abstract

In the first part of this paper [5], the Newman-Schoenberg phenomenon of cardinal logarithmic spline functions is established by means of real transformation methods. In particular, this approach applies an appropriate second-order refinement of Karamata’s Abel-Tauber theorem for the one-sided Laplace transformation. It is the purpose of the second part to investigate the same phenomenon by means of complex transformation methods. The present approach exploits the Pincherle identity, i.e., the inverse Mellin transform of the gamma function G and then applies the calculus of residues of complex function theory. In this way, an asymptotic representation of the error term arises. For details, see the note [6].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. 1.
    De Boor, C: A practical guide to splines. Applied Mathematical Sciences, Vol. 27. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  2. 2.
    Meinardus, G.: Bemerkungen zur Theorie der B-Splines. In: Spline-Funktionen, Vorträge und Aufsätze (Herausgegeben von K. Böhmer, G. Meinardus und W. Schempp), S. 165–175. Mannheim-Wien-Zürich: Bibliographisches Institut 1974Google Scholar
  3. 3.
    Newman, D.J., Schoenberg, I.J.: Splines and the logarithmic function. Pacific J. Math. 61, 241–258 (1975)CrossRefGoogle Scholar
  4. 4.
    Schempp, W.: On the convergence of cardinal logarithmic splines. J. Approx. Theory 23, 108–112 (1978)CrossRefGoogle Scholar
  5. 5.
    Schempp, W.: Approximation und Transformationsmethoden. In: Numerische Methoden der Approximationstheorie, Band 4 (Herausgegeben von L. Collatz, G. Meinardus und H. Werner), S. 299–305. Basel-Stuttgart: Birkhäuser 1978CrossRefGoogle Scholar
  6. 6.
    Scherapp, W.: A note on the Newman-Schoenberg phenomenon. Math. Z. 167, 1–6 (1979)CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • Walter Schempp
    • 1
  1. 1.Lehrstuhl für Mathematik ISiegen 21 BundesrepublikDeutschland

Personalised recommendations