Numerische Mathematik

, Volume 8, Issue 4, pp 392–406

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

Authors

  • Rudolf Gorenflo
    • Institut für Plasmaphysik GmbH
    • Department of MeteorologyThe Hebrew University
  • Yehudith Kovetz
    • Institut für Plasmaphysik GmbH
    • Department of MeteorologyThe Hebrew University
Article

DOI: 10.1007/BF02162982

Cite this article as:
Gorenflo, R. & Kovetz, Y. Numer. Math. (1966) 8: 392. doi:10.1007/BF02162982

Summary

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.

Copyright information

© Springer-Verlag 1966