Summary
The well-known integral transform
arising in spectroscopy, corresponds to half-order differentiation by substitutingr 2 = 1 −s,x 2 = 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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References
Barb, W. L.: Method for computing the radial distribution of emitters in a cylindrical source. Journal Opt. Soc. Am.52, 885–888 (1962).
Doetsch, G.: Einführung in Theorie und Anwendung der Laplace-Transformation. Basel u. Stuttgart: Birkhäuser 1958.
Gorenflo, R.: Numerische Methoden zur Lösung einer Abelschen Integralgleichung. IPP/6/19, Institut für Plasmaphysik, Garching bei München (Germany), Mai 1964.
Künzi, H. P., u.W. Krelle: Nichtlineare Programmierung. Berlin-Göttingen-Heidelberg: Springer 1962.
Tricomi, F. G.: Vorlesungen über Orthogonalreihen. Berlin-Göttingen-Heidelberg: Springer 1955.
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.
— 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.)
—— Lösung einer Abelschen Integralgleichung bei Anwesenheit von Störungen mittels quadratischer Optimierung. Zeitschrift für angewandte Mathematik und Mechanik45, Sonderheft DAMM-Tagung, T33-T35 (1965).
Hohn, F. E.: Elementary matrix algebra. New York: The Macmillan Company 1964.
Becker, L., u.H. W. Drawin: Ein Analogrechengerät zur Lösung der Abelschen Integralgleichung. Zeitschrift für Instrumentenkunde72, 251–256 (1964).
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This work was performed under the terms of the agreement between the Institut für Plasmaphysik GmbH, Munich-Garching, and Euratom to conduct joint research in the field of plasma physics. One of the authors (Gorenflo) talked on this subject at the GSM-conference in Vienna, April 1965, and at the IFIP-Congress in New York, May 1965 (v. [7] and [8]).
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Gorenflo, R., Kovetz, Y. Solution of an Abel-type integral equation in the presence of noise by quadratic programming. Numer. Math. 8, 392–406 (1966). https://doi.org/10.1007/BF02162982
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DOI: https://doi.org/10.1007/BF02162982