Numerische Mathematik

, Volume 8, Issue 4, pp 392–406 | Cite as

Solution of an Abel-type integral equation in the presence of noise by quadratic programming

  • Rudolf Gorenflo
  • Yehudith Kovetz


The well-known integral transform
$$i(r) = - \frac{1}{\pi }\int\limits_{x = r}^1 {\frac{{dI(x)}}{{\sqrt {x^2 - r^2 } }},} 0 \leqq r \leqq 1,I(1) = 0$$
arising in spectroscopy, corresponds to half-order differentiation by substitutingr2 = 1 −s,x2 = 1 − t. Therefore noise is amplified by transforming the measured functionI intoi. Two undesirable effects may arise: (a) lack of smoothness ini (r), (b) intervals in whichi(r) < 0, although for physical reasons we should havei(r) ≧ 0.

After developing a heuristic theory of noise amplification we present a fitting technique for approximate computation ofi(r), using the extra informationi(r) ≧ 0 as a restriction. This leads to a quadratic programming problem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barb, W. L.: Method for computing the radial distribution of emitters in a cylindrical source. Journal Opt. Soc. Am.52, 885–888 (1962).Google Scholar
  2. [2]
    Doetsch, G.: Einführung in Theorie und Anwendung der Laplace-Transformation. Basel u. Stuttgart: Birkhäuser 1958.Google Scholar
  3. [3]
    Gorenflo, R.: Numerische Methoden zur Lösung einer Abelschen Integralgleichung. IPP/6/19, Institut für Plasmaphysik, Garching bei München (Germany), Mai 1964.Google Scholar
  4. [4]
    Künzi, H. P., u.W. Krelle: Nichtlineare Programmierung. Berlin-Göttingen-Heidelberg: Springer 1962.Google Scholar
  5. [5]
    Tricomi, F. G.: Vorlesungen über Orthogonalreihen. Berlin-Göttingen-Heidelberg: Springer 1955.Google Scholar
  6. [6]
    Gorenflo, R., andY. Kovetz: Solution of an Abel type integral equation in the presence of noise. IPP/6/29, Institut für Plasmaphysik, Garching bei München, November 1964.Google Scholar
  7. [7]
    — Computation of an integral transform ofAbel's type in the presence of noise by quadratic programming. Proceedings of IFIP Congress 1965, New York. Volume 2. (In press.)Google Scholar
  8. [8]
    —— Lösung einer Abelschen Integralgleichung bei Anwesenheit von Störungen mittels quadratischer Optimierung. Zeitschrift für angewandte Mathematik und Mechanik45, Sonderheft DAMM-Tagung, T33-T35 (1965).Google Scholar
  9. [9]
    Hohn, F. E.: Elementary matrix algebra. New York: The Macmillan Company 1964.Google Scholar
  10. [10]
    Becker, L., u.H. W. Drawin: Ein Analogrechengerät zur Lösung der Abelschen Integralgleichung. Zeitschrift für Instrumentenkunde72, 251–256 (1964).Google Scholar

Copyright information

© Springer-Verlag 1966

Authors and Affiliations

  • Rudolf Gorenflo
    • 1
    • 2
  • Yehudith Kovetz
    • 1
    • 2
  1. 1.Institut für Plasmaphysik GmbHGarching bei MünchenGermany
  2. 2.Department of MeteorologyThe Hebrew UniversityJerusalemIsrael

Personalised recommendations