Geodesic
Method of Local Peak Functions for Reconstructing the Original Profile in the Fourier Transformation
Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 1, pp. 15-25

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We propose a method for reconstructing the original profile function in the one-dimensional Fourier transformation from the module of the Fourier transform function analytically. The major concept of the method consists in representing the modeling profile function as a sum of local peak functions. The latter are chosen as eigenfunctions generated by linear differential equations with polynomial coefficients. This allows directly inverting the Fourier transformation without numerical integration. The solution of the inverse problem thus reduces to a nonlinear regression with a small number of optimizing parameters and to a numerical or asymptotic study of the corresponding modeling peak functions taken as the eigenfunctions of the differential equations and their Fourier transforms. 
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     author = {X. Dosch and S. Yu. Slavyanov},
     title = {Method of {Local} {Peak} {Functions} for {Reconstructing} the {Original} {Profile} in the {Fourier} {Transformation}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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X. Dosch; S. Yu. Slavyanov. Method of Local Peak Functions for Reconstructing the Original Profile in the Fourier Transformation. Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 1, pp. 15-25. http://geodesic.mathdoc.fr/item/TMF_2002_131_1_a2/