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
Article

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.

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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

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